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Subconstituent algebras of Latin squares

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Subconstituent algebras of Latin squares
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Daqqa, Ibtisam
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Terwilliger Algebra
Strongly regular graph
Association scheme
Bose -Mesner Algebra
Fusions
Dissertations, Academic -- Mathematics -- Doctoral -- USF   ( lcsh )
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Abstract:
ABSTRACT: It is well-known that one may construct a 4-class association scheme on the positions of a Latin square, where the relations are the identity, being in the same row, being in the same column, having the same entry, and everything else. We describe the subconstituent (Terwilliger) algebras of such an association scheme. One also may construct several strongly regular graphs on the positions of a Latin square, where adjacency corresponds to any subset of the nonidentity relations described above. We describe the local spectrum and subconstituent algebras of such strongly regular graphs. Finally, we study various notions of isomorphism for subconstituent algebras using Latin squares as examples.
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Dissertation (Ph.D.)--University of South Florida, 2008.
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by Ibtisam Daqqa.
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SubconstituentAlgebrasofLatinSquaresbyIbtisamDaqqaAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofMathematicsCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:BrianCurtin,Ph.D.MasahikoSaito,Ph.D.Xiang-dongHou,Ph.D.BrendanT.Nagle,Ph.D.DateofApproval:November29,2007Keywords:Terwilligeralgebra,Bose-Mesneralgebra,Associationscheme,Stronglyregulargraph,FusionscCopyright2008,IbtisamDaqqa

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DedicationTomylovingparents,mysupportivehusband,andmythreelovelydaughters,Leen,Dana,andYasmeen.

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AcknowledgementsIwouldliketogratefullyandsincerelythankDr.BrianCurtinforhisguidance,understanding,patience,andkindness.Hismentorshipwasparamountinprovidingawell-roundedexperienceconsistentwithmylong-termcareergoals.Foreverythingyou'vedoneforme,Dr.Brian,Ithankyou.IwouldalsoliketothanktheDepartmentofMathematicsatUniversityofSouthFlorida,especiallythosemembersofmydoctoralcommitteefortheirinput,valuablediscussionsandaccessibility.Manythanksgototheformerandcurrentadministrativestainthedepartment,Beverly,Nancy,andMaryAnnfortheirkindnessandassistance.Iwouldliketothankmanyfriendswhohavehelpedmestaysanethroughthesedicultyears,especiallyDr.Kheira,andDr.Ibrahimouandhiswife.IgreatlyvaluetheirfriendshipandIdeeplyappreciatetheirhelp.I'malsogratefultoDr.HassanAlnajjarandhiswifewhohelpedmeadjusttoanewcountry.Finally,andmostimportantly,IwouldliketothankmyhusbandNourAldeen.Hissupport,encouragement,quietpatienceandunwaveringlovewereundeniablythebedrockuponwhichthepasttenyearsofmylifehavebeenbuilt.Ithankmyparents,fortheirfaithinmeandallowingmetobeasambitiousasIwanted.Ialsothankmybrothersandmysistersfortheirendlesshelpandsupport.AboveallImustthankAllah"whoencouragedandcomfortedmeinthewholeprocessofthisdissertation.

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TableofContentsListofTablesiiiListofFiguresivAbstractv1Introduction12AlgebraicPreliminaries52.1TheBose-MesnerAlgebra........................ 5 2.2TheDualBose-MesnerAlgebra..................... 6 2.3TheSubconstituentAlgebraanditsModules............. 7 3TheSubconstituentAlgebraofaLatinSquare113.1LatinSquaresandBose-MesnerAlgebras............... 11 3.2SomePermutations........................... 14 3.3CycleModules.............................. 16 3.4DecompositionintoIrreducibleModules................ 20 3.5SomeIntermediateModules....................... 28 3.6CollectingCycleModules........................ 30 3.7TheFourthSubconstituent....................... 36 3.8OtherResults.............................. 39 3.9CayleyTablesofFiniteGroups..................... 39 3.10SmallLatinSquares........................... 40 i

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4StronglyRegularGraphsfromaLatinSquare434.1StronglyRegularGraphs........................ 43 4.2StronglyRegularGraphsfromaLatinSquare............. 45 4.3Fusions.................................. 46 4.4G3..................................... 48 4.5G2..................................... 53 4.6G1..................................... 54 5Isomorphisms565.1IsomorphismsofBose-MesnerAlgebra................. 56 5.2IsomorphismsofSubconstituentAlgebras............... 58 5.3EquivalencesofLatinSquares...................... 63 5.4IsomorphismsandLatinSquares.................... 66 References72Appendices80AppendixA:PermittedRootsofUnity...................... 81 AppendixB:ComputerCode........................... 90 AbouttheAuthorEndPage ii

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ListofTables5.1ThenumbersofLatinsquaresofvarioussizes............... 65 5.2EquivalenceclassesofLatinsquares..................... 65 5.3Numberofpossiblestructuralisomorphismclasses............. 68 5.4NumberofpossibleBose-Mesnerisomorphismclasses........... 68 5.5Numberofpossibleabstractisomorphismclasses............. 69 6IrreducibleT-modulesforn=5....................... 81 7IrreducibleT-modulesforn=6....................... 82 8IrreducibleT-modulesforn=7....................... 83 9IrreducibleT-modulesforn=8....................... 84 10IrreducibleT-modulesforn=9,pt.1................... 85 11IrreducibleT-modulesforn=9,pt.2................... 86 12IrreducibleT-modulesforn=10,pt.1................... 87 13IrreducibleT-modulesforn=10,pt.2................... 88 14IrreducibleT-modulesforn=10,pt.3................... 89 iii

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ListofFigures1.1TwoCayleytables.............................. 1 1.2Twopointsequenceswithrespectto;1;1................. 3 3.1TheinterleavingofcyclesLemma3.2.5,Denition3.2.7........ 16 3.2Theactiononu1;i,u2;i,u3;i2WC1;C2;C3................. 19 3.3Theactiononv1;i2WC1;C2;C3...................... 19 3.4Theactiononv2;i2WC1;C2;C3...................... 19 3.5Theactiononv3;i2WC1;C2;C3...................... 20 3.6Theactiononui2WC1;C2;C3...................... 26 3.7Theactiononv12WC1;C2;C3...................... 26 3.8Theactiononv22WC1;C2;C3...................... 26 3.9Theactiononv32WC1;C2;C3...................... 26 iv

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SubconstituentalgebraofLatinSquaresIbtisamDaqqaAbstractLetnbeapositiveinteger.ALatinsquareofordernisannnarrayLsuchthateachelementofsomen-setoccursineachrowandineachcolumnofLexactlyonce.Itiswell-knownthatonemayconstructa4-classassociationschemeonthepositionsofaLatinsquare,wheretherelationsaretheidentity,beinginthesamerow,beinginthesamecolumn,havingthesameentry,andeverythingelse.WedescribethesubconstituentTerwilligeralgebrasofsuchanassociationscheme.OnealsomayconstructseveralstronglyregulargraphsonthepositionsofaLatinsquare,whereadjacencycorrespondstoanysubsetofthenonidentityrelationsdescribedabove.Wedescribethelocalspectrumandsubconstituentalgebrasofsuchstronglyregulargraphs.Finally,westudyvariousnotionsofisomorphismforsubconstituentalgebrasusingLatinsquaresasexamples. v

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1IntroductionLetnbeapositiveinteger.ALatinsquareofordernisannnarrayLsuchthateachelementofsomen-setoccursineachrowandineachcolumnofLexactlyonce.ThesimplestexamplesaretheCayleytableofagroup,asinFigure1.1.Sudokupuzzles,whencompleted,formaLatinsquare. 0BBBB@12345234513451245123512341CCCCA0BB@12342143341243211CCAZ5Z2Z2 Figure1.1:TwoCayleytables LatinsquareswereintroducedbyEulerin1779inthecontextofapuzzleinrecre-ationalmathematics.SincethattimeLatinsquareshavefoundapplicationsinavarietyofmathematicalareas.Sucheldsincludegrouptheory,graphtheory,nitegeome-tries,codingtheory,designtheory,cryptography,andstatistics.Theseconnectionsaredevelopedinthegeneralreferences[42,43,69].LetLdenoteaLatinsquareofordern3,andencodeLwiththesetX=fi;j;Li;jj1i;jng.DeneverelationsonX:Forallx=i;j;Li;j,x0=i0;j0;Li0;j02X,R0identity:xR0x0ifx=x0,R1samerow:xR1x0ifi=i0andx6=x0, 1

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R2samecolumn:xR2x0ifj=j0andx6=x0,R3sameentry:xR3x0ifLi;j=Li0;j0andx6=x0,R4everythingelse:xR4x0ifi6=i0,j6=j0,andLi;j6=Li0;j0.Itiswell-knownthatX;fRig4i=0isasymmetricassociationscheme[5,41],whichwerefertoastheassociationschemeofL.HencethecharacteristicmatricesoftheverelationscomprisethebasisofHadamardidempotentsofaBose-MesneralgebraM,whichwerefertoastheBose-MesneralgebraofLsee[5].Bose-Mesneralgebrasrstaroseinstatisticaldesigns[15,16],incentralizeralgebrasofpermutationgroups[90],andinconnectionwithdistance-transitivegraphs[12].Aperiodofgrowthinthesubjectoccurredinthe1980'safterDelsartedemonstratedapplicationstocodesanddesigns[41]andastheclassicationofnitesimplegroupsmotivatedandaidedthestudyofdistance-transitivegraphs.Detailscanbefoundin[5,10,13,17,51].Bose-Mesneralgebrashavebeengeneralizedinseveraldirections,suchascoherentalgebras[53,54],tablealgebras[1,2,3,4],group-likeassociationschemes[91].Morerecently,connectionshavebeendevelopedtolinkinvariants[22,61,62],quantumgroups[38,60],fusionalgebras[8],maximalabeliansubalgebras[52,75],commutingsquares[6],andsubfactors[31,63,64].InthisworkwestudythesubconsitutentorTerwilligeralgebrasoftheassociationschemeofaLatinsquare.ThesubconstituentalgebrarenestheBose-Mesneralgebrabyencodingadditionalcombinatorialinformationconcerningtherelationbetweeneachpointandthexedbasepoint.ItisknownthatthealgebraicisomorphismclassoftheBose-MesneralgebraofaLatinsquaredependsonlyuponitsorderandnootherpropertyoftheLatinsquare.OneofourmotivationsistobetterdistinguishLatinsquaresusingsubconstituentalgebras.Subconstitutentalgebrashavebeenstudiedinseveralpapers.Themesofthesepapersincludecompletedescriptionofthesubconstituentalgebraofsomeclassofassociationscheme[7,11,19,20,21,30,44,49,83,88],partialdescriptionofthesubconstituentalgebraofsomeclassofassociationscheme[23,24,29,36,45,55,58,59,84,86,87],descriptionsoftheassociationschemeswhosesubconstituentalgebrasatisessomecon-dition[25,39],correspondencesbetweencertaincombinatorialoftenlocalandalgebraic 2

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conditions[27,32,39,50,72,73],algebraicconnections[33,34,38,89,84],andgeneralizations[46,47,60,81].Fixabasepointp=rp;cp;ep2X.LetTdenotethesubconstituentalgebraoftheassociationschemeofLwithrespecttop.WedescribetheactionontheirreducibleT-modulesintermsofasubstructureoftheLatinsquarewhichwenowdescribe.BythedenitionofaLatinsquareanytwocoordinatesofanelementofXuniquelydeterminethethird:WerefertothisastheLatinsquareproperty.Formasequenceofpointsasfollows.Pickx12Xsuchthatx1R1p{writex1=rp;c1;e1.Oncex1ischosen,allsubsequentpointsareuniquelydeterminedbytheLatinsquareproperty.Givenxi=rp;ci;ei,letyi2XbetheuniquepointsuchthatyiR2pandyiR3xi{writeyi=ri;cp;ei.Letzibetheuniquezi2XsuchthatziR3pandziR1yi{writezi=ri;ci+1;ep.Finally,letxi+12Xbetheuniquepointsuchthatxi+1R1pandxi+1R2zi{writexi+1=rp;ci+1;ei+1.Repeatthisprocessuntilxk+1=x1.Viewx1,x2,...,xkasacycleinapermutationofthen)]TJ/F19 11.955 Tf 12.298 0 Td[(1pointsinrelationR1withp,wheretheothercyclesconstructedsimilarly.Thepermutationconstructeddependsuponthebasepointp.Figure1.2interpretstwosequencesofpointsinaLatinsquarewithrespectto;1;1:x1=;2;2,y1=;1;2,z1=;3;1,x2=;3;3,y2=;1;3,z2=;2;1andx01=;4;4,y01=;1;4,z01=;4;1.Thisgivesrisetoapermutationwithone1-cycleandone2-cycle,namelyx1x2x01. 1234241331424321.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. Figure1.2:Twopointsequenceswithrespectto;1;1. InChapter3,wedescribetheirreducibleT-modules.Wesummarizetheresultsnow.Thereisalwaysaunique5-dimensionalirreducibleT-module,andtherearen2)]TJ/F19 11.955 Tf 12.327 0 Td[(6n+7manymutuallyisomorphic1-dimensionalT-modulesinthestandardmodule.The 3

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remainingT-modulesarerelatedtothecyclesofthepermutationconstructedabove.Ifthereisjustonecyclewithk=n)]TJ/F19 11.955 Tf 12.918 0 Td[(1,thenthereisa6-dimensionalirreducibleT-moduleassociatedwitheachkthrootofunityexcept1itself.Ifthereismorethanonecycle,thenforeachcycleoflengthkthereisa6-dimensionalirreducibleT-moduleassociatedwitheachkthrootofunity.However,inthiscaseoneoftheirreducibleT-modulesassociatedwith1mustbedroppedtoobtainlinearindependence.TheT-actiononeachofthesen)]TJ/F19 11.955 Tf 10.556 0 Td[(2-many6-dimensionalirreducibleT-modulesisentirelydeterminedbytheassociatedrootofunity:Twosuchmodulesareisomorphicifandonlyiftheyareassociatedwiththesamerootofunity.OnemaydeneastronglyregulargraphwithvertexsetXforeachsubsetCofcoordinates1,2,3bydeclaringdistinctx=i;j;Li;jandx0=i0;j0;Li0;j0tobeadjacentifforsomecoordinatec2C,xc=x0c.InChapter4wedescribethelocalspectrumandsubconstituentalgebraofthesestronglyregulargraphsusingthecyclesdescribedabove.ThekeyideaisthattheirreducibleT-modulesremainmodulesnolongerirreducibleforthesubconstituentalgebrasofthesestronglyregulargraphs.ThusweshowhowtodecomposethesealreadysmallirreducibleT-modulesintoirreduciblemodulesforthestronglyregulargraphs'subconstituentalgebras.Thenthelocalspectrumisdeterminedbyactingonthesemodulesbyanappropriatematrix.InChapter5,wediscussisomorphismsofsubconstituentalgebras.ThesubconstituentalgebraofaLatinsquarewithrespecttosomebasepointisisomorphictoM5M`6M1,where`isthenumberofmutuallynonisomorphicirreducibleT-modulesofdimension6.Manydistinctcyclestructuresmaygiverisetothesamevalueof`.However,theactionofthesubconstituentalgebraoneachirreducibleT-moduleisuniquelydeterminedbythecyclestructureofthepermutationconstructedabove.InAppendixAwegivestablesrelatingcyclestructureinsmallexamplestopossi-bleirreduciblemodules,andinAppendixBwegiveMathematicacodewhichproducestheirreduciblemodulesofthesubconstituentalgebraofaLatinsquareandtherelatedstronglyregulargraphs. 4

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2AlgebraicPreliminaries 2.1TheBose-MesnerAlgebraInthissectionwerecallsomebackgroundmaterial.WerstrecallBose-Mesneralgebrasandsomeoftheirbasicproperties.Generalreferencesforthesubjectinclude[10,17,51].LetXdenoteanite,nonemptyset,andletMXdenotethecomplexalgebraofmatriceswithcomplexentrieswhoserowsandcolumnsareindexedbyX.ForA2MXandforx,y2X,letAx;ydenotethex;y-entryofA.ForA,B2MX,letABdenotetheHadamardproductofAandB:ABx;y=Ax;yBx;y.TheordinarymatrixproductofAandBwillbedenotedbyjuxtaposition:AB.ForA2MX,lettAdenotethetransposeofA.ABose-MesneralgebraonXisacommutativesubalgebraofMXwhichisclosedunderHadamardproduct,whichisclosedundertransposition,andwhichcontainstheidentitymatrixIandtheall-onesmatrixJ.LetMdenotead+1-dimensionalBose-MesneralgebraonX.ThebasisofHadamardidempotentsofMistheuniquebasisfAigdi=0suchthatA0=I; .1.1 AiAj=ijAii;jd; .1.2 dXi=0Ai=J; .1.3 whereijdenotestheKroneckersymbol.LetA0,A1,...,AdbeanorderingoftheHadamardidempotentsofM.Thenrelativetothisordering,theintersectionnumbers 5

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phijofMaredenedbyAiAj=Xdh=0phijAhi;jd: .1.4 ThebasisofprimitiveidempotentsofMistheuniquebasisfEigdi=0suchthatE0=n)]TJ/F25 7.97 Tf 6.587 0 Td[(1J; .1.5 EiEj=ijEi; .1.6 dXi=0Ei=I: .1.7 LetE0,E1,...,EdbeanorderingoftheprimitiveidempotentsofM.Thenrelativetothisordering,theKreinparametersqhijofMaredenedbyEiEj=n)]TJ/F25 7.97 Tf 6.586 0 Td[(1Xdh=0qhijEhi;jd: .1.8 RelativetotheorderingsoftheHadamardandprimitiveidempotents,theeigenvaluesPj;iandthedualeigenvaluesQj;iofMaredenedrespectivelybyAi=Xdj=0Pj;iEjid; .1.9 Ei=n)]TJ/F25 7.97 Tf 6.586 0 Td[(1Xdj=0Qj;iAjid: .1.10 Thed+1d+1matricesPandQwithj;i-entriesPj;iandQj;iarecalledtheeigenmatrixanddualeigenmatrixofM,respectively.Itisknownthattheeachofthesesetsofparameterstheintersectionnumbers,theKreinparameters,theeigenmatrix,andthedualeigenmatrixdeterminestheothers.See,forexample,[10]forprecisedetails. 2.2TheDualBose-MesnerAlgebraWenowrecallfrom[84]thedualBose-MesneralgebraofaBose-MesneralgebraMonX.Fixp2X.ForeachA2M,letA2MXdenotethediagonalmatrixwithx;x-entryAx;x=Ap;xx2X.LetM=M.WerefertoMasthe 6

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dualBose-MesneralgebraofMwithrespecttop.Observethat:M!MisalinearbijectionandAB=AB.SetEi=Ai.ThenfEigdi=0isabasisofM.WerefertofEigdi=0asthebasisofdualidempotents.Applyingto.1.2and.1.3givesEiEj=ijEii;jd; .2.11 dXi=0Ei=I: .2.12 SetAi=nEi0id.ThenfAigdi=0isabasisofM.WerefertofAigdi=0asthebasisofdualHadamardidempotentsofM.Applyingto.1.5,.1.7,and.1.8givesA0=I; .2.13 dXi=0Ai=nE0; .2.14 AiAj=Xdh=0qhijAh;i;jd: .2.15 Applyingto.1.9and.1.10givesAi=Xdj=0Qj;iEjid; .2.16 Ei=n)]TJ/F25 7.97 Tf 6.586 0 Td[(1Xdj=0Pj;iAjid: .2.17 2.3TheSubconstituentAlgebraanditsModulesWerecallfrom[84]somefactsconcerningthesubconstituentalgebraofaBose-MesneralgebraMonX.Fixp2X.ThesubconstituentorTerwilligeralgebraofMwithrespecttopisthesubalgebraofMXgeneratedbyM[M.By.1.1,.1.4,2.2.11,and.2.12,TisalsogeneratedbyfEiAjEkj0i;j;kdg. Lemma2.3.1 [84]Forallh,i,jh;i;jd,EiAhEj=0ifandonlyifphij=0: .3.18 7

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Theorem2.3.2 [84]Subconstituentalgebrasaresemisimple.WemayappealtoWedderburntheory[40]todescribesubconstituentalgebrasbytheirirreduciblemodules.LetV=CXdenotethecolumnvectorspacewithentriesindexedbyX.EndowVwiththeHermitianinnerproductdenedbyhu;vi=tuv.ObservethatMXactsonVbyleft-multiplication.WerefertoVasthestandardmoduleforT.ByaT-modulewemeanalinearsubspaceUofVwhichisclosedundertheactionofT:Au2UforallA2Tandforallu2U.LetbeanindexsetfortheisomorphismclassesofirreducibleT-modules.LetVbethesumofallirreducibleT-modulesintheisomorphismclassofirreducibleT-modulesindexedby2.EachVisanorthogonaldirectsumofmutuallyisomorphicirreducibleT-modules.WhilethedirectsummandsofVarenotnecessarilyunique,theirnumberis.Writemulttodenotethisnumber,andforanyirreducibleT-moduleWcontainedinV,setmultW=mult.WerefertomultWasthemultiplicityofW.NotethatV=2Vorthogonaldirectsum.Wenotethefollowingconsequenceofthisdiscussion. Lemma2.3.3 SupposeWandW0arenon-isomorphicirreducibleT-modules.ThenWandW0areorthogonaltooneanother.TheprimitivecentralidempotentsofTarealsoindexedbyastheyareinbijectivecorrespondencewiththeV.ForeachsubspaceVthereisauniqueprimitivecentralidempotent'suchthatV='V,andconverselyforeachprimitivecentralidempotent','V=Vforsome2.LetWVbeanirreducibleT-module.ThenthemaptakingL2'Ttotheendomorphismw7!Lww2Wisanisomorphism.Thus'T=EndCW.However,EndCWisisomorphictothekkcomplexmatrixalgebra,wherekisthedimensionofW.NowT=2'Tdirectsum.ThusTisisomorphictoadirectsumofcomplexmatrixalgebras.ForallelementsxofX,dene[[x]]2Vtobethecharacteristicvectorofx,ie,thevectorwithaoneintherowindexedbyxandzeroseverywhereelse.Observethatthesetf[[x]]jx2XgisthestandardbasisofV.IfSisamulti-setofelementsofX,dene 8

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[[S]]=Px2S[[x]],whereeachsummandoccursonceforeachoccurrenceinS.Forallx2X,dene)]TJ/F28 7.97 Tf 48.285 -1.793 Td[(ix=fy2XjxRiyg. Lemma2.3.4 [84]Forallx2X,Ei[[x]]=8><>:[[x]]ifpRix;0otherwise;Ai[[x]]=[[)]TJ/F28 7.97 Tf 45.366 -1.793 Td[(ix]]:Inparticular,EiAjEk[[x]]=8><>:[[)]TJ/F28 7.97 Tf 11.825 -1.793 Td[(ip)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(jx]]ifxRkp;0otherwise.Proof.FromthedenitionofAi,0id,wehaveAi[[x]]=PyRix[[y]]=[[)]TJ/F28 7.97 Tf 35.912 -1.793 Td[(ix]].NowsinceEiisdiagonalwhereEiy;y=1ifyRipandzeroanywhereelse,wehaveEi[[x]]=hi[[x]],wherehsuchthatpRhx.ForEiAjEk,wehaveEiAjEk[[x]]=8><>:EiAj[[x]]ifxRkp,0otherwise,=8><>:[[)]TJ/F28 7.97 Tf 11.825 -1.794 Td[(jx)]TJ/F28 7.97 Tf 7.314 -1.794 Td[(ip]]ifxRkp,0otherwise.2Thesupportofavectorv2Visthesetsuppv=fx2Xjvx6=0g. Lemma2.3.5 LetUVbeasubsetofnonzerovectors.SupposethatforallsubsetsSUandallu2UnS,thesymmetricdierenceof[s2Ssuppsandsuppuisnonempty.ThenUislinearlyindependent.Proof.SupposethatPki=1iui=0,whereui2U.LetS=fu2;u3;:::;ukgandu=u1.Thenbyassumptionthesymmetricdierenceof[s2Ssuppsandsuppuisnonempty.Inparticular,thesumisnonzerounless1=0.Proceedingbyinduction,wendthati=0foralli,soUislinearlyindependent.2 9

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WerecallsomefactsaboutaspecialirreducibleT-module. Lemma2.3.6 [84]Foralli,j,ki;j;kdEjAkEi[[)]TJ/F28 7.97 Tf 11.825 -1.793 Td[(ip]]=pkij[[)]TJ/F28 7.97 Tf 11.825 -1.793 Td[(jp]]: Lemma2.3.7 [84]Foralli,j,ki;j;kdandforallx2XEiJEk[[x]]=8><>:[[)]TJ/F28 7.97 Tf 11.825 -1.793 Td[(ip]]ifxRkp,0otherwise.Proof.ByLemma2.3.4,EjAkEi[[)]TJ/F28 7.97 Tf 11.825 -1.793 Td[(ip]]=XxRipEjAkEi[[x]]=XxRip[[)]TJ/F28 7.97 Tf 11.825 -1.793 Td[(kx)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(jp]]=XxRipyRkxyRjppkij[[y]]=pkij[[)]TJ/F28 7.97 Tf 11.825 -1.793 Td[(jp]]:2 Theorem2.3.8 [84]ThereisanirreducibleT-modulewithbasisf[[)]TJ/F28 7.97 Tf 11.825 -1.793 Td[(ip]]j0idg.WerefertoastheprimaryT-moduleanddenotebyP.Proof.Observethatf[[)]TJ/F28 7.97 Tf 11.825 -1.793 Td[(ip]]j0idgspansaT-modulebyLemma2.3.6.Thislemmaalsoimpliesthatthismoduleisirreducible.ThesevectorsarelinearlyindependentbyLemma2.3.5,sotheresultfollows.2 10

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3TheSubconstituentAlgebraofaLatinSquare 3.1LatinSquaresandBose-MesnerAlgebrasLetLdenoteaLatinsquareofordern3.EncodeLwiththesetX=fi;j;Li;jj1i;jng.ThisencodingissometimesreferredtoastheorthogonalarrayrepresentationofL.TodescribetheassociatedBose-MesneralgebrawerstdeneverelationsonXasfollows:Forallx=i;j;Li;j,x0=i0;j0;Li0;j02X,R0identity:xR0x0ifx=x0, .1.1 R1samerow:xR1x0ifi=i0andx6=x0, .1.2 R2samecolumn:xR2x0ifj=j0andx6=x0, .1.3 R3sameentry:xR3x0ifLi;j=Li0;j0andx6=x0, .1.4 R4everythingelse:xR4x0ifi6=i0,j6=j0,andLi;j6=Li0;j0. .1.5 NowX;fRig4i=0isacommutativeassociationscheme,andhencethecharacteristicmatricesoftheverelationscomprisethebasisofHadamardidempotentsofaBose-MesneralgebraM,whichwerefertoastheBose-MesneralgebraofLsee[5].TheequivalenceofBose-Mesneralgebrasandcommutativeassociationschemesiswell-known[10,17].GivenaLatinsquareLanditsassociationschemeX;fRigdi=0,denefori=0,1,2,3,4,matricesAi2MXbyAix;y=1ifxRiyand0otherwisex;y2X.ThenfAig4i=0isthebasisofHadamardidempotentsofaBose-Mesneralgebra,whichwerefertoastheBose-MesneralgebraofL.WenowrecallsomefactsconcerningtheBose-MesneralgebraofaLatinsquare.Forcompletenessweprovidebriefproofsofthesefacts.Theirintersectionnumbersarewell-known. 11

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Theorem3.1.1 [5]LetLdenoteaLatinsquareofordern,andletMdenotetheBose-MesneralgebraofM. i TheintersectionnumbersofMaregivenbyp0ij=0BBBBBBBB@100000n)]TJ/F19 11.955 Tf 11.955 0 Td[(100000n)]TJ/F19 11.955 Tf 11.955 0 Td[(100000n)]TJ/F19 11.955 Tf 11.955 0 Td[(100000n2)]TJ/F19 11.955 Tf 11.955 0 Td[(3n+21CCCCCCCCA;p1ij=0BBBBBBBB@010001n)]TJ/F19 11.955 Tf 11.955 0 Td[(20000001n)]TJ/F19 11.955 Tf 11.955 0 Td[(20010n)]TJ/F19 11.955 Tf 11.955 0 Td[(200n)]TJ/F19 11.955 Tf 11.955 0 Td[(2n)]TJ/F19 11.955 Tf 11.955 0 Td[(2n2)]TJ/F19 11.955 Tf 11.955 0 Td[(5n+61CCCCCCCCA;p2ij=0BBBBBBBB@001000001n)]TJ/F19 11.955 Tf 11.955 0 Td[(210n)]TJ/F19 11.955 Tf 11.955 0 Td[(2000100n)]TJ/F19 11.955 Tf 11.955 0 Td[(20n)]TJ/F19 11.955 Tf 11.955 0 Td[(20n)]TJ/F19 11.955 Tf 11.955 0 Td[(2n2)]TJ/F19 11.955 Tf 11.955 0 Td[(5n+61CCCCCCCCA;p3ij=0BBBBBBBB@000100010n)]TJ/F19 11.955 Tf 11.955 0 Td[(20100n)]TJ/F19 11.955 Tf 11.955 0 Td[(2100n)]TJ/F19 11.955 Tf 11.955 0 Td[(200n)]TJ/F19 11.955 Tf 11.955 0 Td[(2n)]TJ/F19 11.955 Tf 11.955 0 Td[(20n2)]TJ/F19 11.955 Tf 11.955 0 Td[(5n+61CCCCCCCCA; 12

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p4ij=0BBBBBBBB@000010011n)]TJ/F19 11.955 Tf 11.955 0 Td[(30101n)]TJ/F19 11.955 Tf 11.955 0 Td[(30110n)]TJ/F19 11.955 Tf 11.955 0 Td[(31n)]TJ/F19 11.955 Tf 11.955 0 Td[(3n)]TJ/F19 11.955 Tf 11.955 0 Td[(3n)]TJ/F19 11.955 Tf 11.955 0 Td[(3n2)]TJ/F19 11.955 Tf 11.955 0 Td[(6n+101CCCCCCCCA:Proof.LetAr=A1+I,Ac=A2+I,andAe=A3+IwhereA1,A2,A3aretherespectiveadjacencymatricesoftherelationsR1,R2,R3onX.ThenAristhematrixoftheunionofrelationsR0andR1,etc.TocomputepkijwewillcomputeAiAjfor0i;j3.Todoso,letRr=R0[R1,Rc=R0[R2,Re=R0[R3.NotethatArArx;y=X2XArx;Ar;y=jf:Rrx;Rrygj=nArx;y:i.eArAr=nAr.SimilarlyA2c=nAcandA2e=nAe.AlsonotethatArAcx;y=jf:Rrx;Rcygj=1:ThusArAc=J.SimilarlyAcAr;ArAe;AeAr;AcAe,andAeAcareequaltoJ.Nowusingtheabovecomputationswecomputephijforij,0i;j3and0h4:A1A1=Ar)]TJ/F27 11.955 Tf 11.955 0 Td[(IAr)]TJ/F27 11.955 Tf 11.955 0 Td[(I=A2r)]TJ/F19 11.955 Tf 11.955 0 Td[(2Ar+I=n)]TJ/F19 11.955 Tf 11.955 0 Td[(2Ar)]TJ/F27 11.955 Tf 11.955 0 Td[(I+n)]TJ/F19 11.955 Tf 11.955 0 Td[(1I=n)]TJ/F19 11.955 Tf 11.955 0 Td[(2A1+n)]TJ/F19 11.955 Tf 11.955 0 Td[(1A0:Thusp011=n)]TJ/F19 11.955 Tf 10.448 0 Td[(1,p111=n)]TJ/F19 11.955 Tf 10.448 0 Td[(2,andp211=p311=p411=0.Similarlywecomputeallotherintersectionnumbersphij,for0i;j3,ij,and0h4.Usingthesymmetrywededucephij,for0i;j3,i
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3.2SomePermutationsWenowbeginourstudyofthesubconstituentalgebraoftheBose-MesneralgebraofaLatinsquare.Inthissectionweshowthatcertainelementsofthesubcontsituentalgebrainducepermutationson)]TJ/F25 7.97 Tf 132.243 -1.794 Td[(1p,on)]TJ/F25 7.97 Tf 35.279 -1.794 Td[(2p,andon)]TJ/F25 7.97 Tf 58.04 -1.794 Td[(3p. Notation3.2.1 LetLdenoteaLatinsquareofordern3andwithsymbolsetf1;2;:::;ng.LetXdenotethesetfi;j;Li;jj1i;jng.LetMdenotetheBose-MesneralgebraofL.Fixp=rp;cp;ep2X,andletTdenotethesubconstitutentalgebraofMwithrespecttop. Lemma3.2.2 WithNotation3.2.1,xapermutationi,j,kof1,2,3.Foreachx2)]TJ/F28 7.97 Tf 7.315 -1.793 Td[(ip,therowofEiAjEkindexedbyxhasauniqueentryequaltooneandallotherentriesareequaltozero.Foreachy2)]TJ/F28 7.97 Tf 7.315 -1.793 Td[(kp,thecolumnofEiAjEkindexedbyyhasauniqueentryequaltooneandallotherentriesareequaltozero.AllotherentriesofEiAjEkarezero.Proof.ImmediatefromthedenitionsofEi,Aj,andEk,andthefactthatpjik=1byTheorem3.1.1.2WenotetheactionofEiAjEkonthestandardbasisofthestandardmoduleintheLatinsquarecase. Lemma3.2.3 WithNotation3.2.1,xapermutationi,j,kof1,2,3.Letx2X.ThenEiAjEk[[x]]=xk;pk[[y]],whereyi=pi,yj=xj,andykisuniquelydeterminedbytheLatinsquareproperty.Proof.ThesuminLemma2.3.4runsoverallysuchthatyi=piandyj=xjbythedenitionsoftherelations.ThereisexactlyonesuchybytheLatinsquareproperty.2 Lemma3.2.4 WithNotation3.2.1,forallpermutationsi,j,kof1,2,3,theprincipalminorofEiAjEkAiEjAkEiindexedby)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(ipisapermutationmatrixandeveryotherentryofEiAjEkAiEjAkEiiszero. 14

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Proof.InlightofLemma5.3.2,itisenoughtotreatjustE1A2E3A1E2A3E1actingonanelementof)]TJ/F25 7.97 Tf 80.152 -1.793 Td[(1p.ByLemma3.2.3,forallrp;c;e2)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(1pE1A2E3A1E2A3E1[[rp;c;e]]=E1A2E3A1E2[[r;cp;e]]=E1A2E3[[r;c0;ep]]=[[rp;c0;e0]];wherec0ande0areuniquelydeterminedbytheLatinsquareproperty.Notethatrp;c0;e02)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(1p.ThustheprincipalminorofE1A2E3A1E2A3E1indexedby)]TJ/F25 7.97 Tf 65.801 -1.793 Td[(1pisapermutationmatrix.AllotherentriesarezerobyLemma3.2.2.2 Lemma3.2.5 WithreferencetoLemma3.2.4,thefollowingareequivalent. i E1A2E3A1E2A3E1inducesak-cycleon)]TJ/F25 7.97 Tf 7.314 -1.794 Td[(1poftheformrp;c1;e1,rp;c2;e2,...,rp;ck;ek. ii E2A3E1A2E3A1E2inducesak-cycleon)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(2poftheformr1;cp;e1,r2;cp;e2,...,rk;cp;ek. iii E3A1E2A3E1A2E3inducesak-cycleon)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(3poftheformr1;c2;ep,r2;c3;ep,...,rk;c1;ep.Proof.iii:ByLemma3.2.3,E2A3E1[[rp;ci;ei]]=[[ri;cp;ei]];E2A3E1[[rp;ci+1;ei+1]]=[[ri+1;cp;ei+1]]:Byi,E1A2E3A1E2A3E1[[rp;ci;ei]]=[[rp;ci+1;ei+1]].HenceE2A3E1A2E3A1E2[[ri;cp;ei]]=E2A3E1E1A2E3A1E2A3E1[[rp;ci;ei]]=E2A3E1[[rp;ci+1;ei+1]]=[[ri+1;cp;ei+1]]: 15

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[[ri)]TJ/F25 7.97 Tf 6.586 0 Td[(1;ci;ep]].................................................................321.................................................................123[[rp;ci;ei]].................................................................231.................................................................132[[ri;cp;ei]].................................................................312.................................................................213[[ri;ci+1;ep]].................................................................321.................................................................123[[rp;ci+1;ei+1]] Figure3.1:TheinterleavingofcyclesLemma3.2.5,Denition3.2.7 Thusiimpliesii.Theotherconsequencesareproveninasimilarmanner.SeeFigure3.1,whereweabbreviateijk=EiAjEk.2 Corollary3.2.6 WithreferencetoLemma3.2.4,thecyclestructureofthepermutationon)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(ipinducedbyEiAjEkAiEjAkEiisindependentofi,j,kforallpermutationsi,j,kof1,2,3.WerefertothecommoncyclestructureasthecyclestructureofLwithrespecttop.Proof.ClearfromLemma3.2.5.2 Denition3.2.7 WithreferencetoLemma3.2.5,werefertothetripleofk-cyclesC1=rp;c1;e1,rp;c2;e2,...,rp;ck;ek;C2=r1;cp;e1,r2;cp;e2,...,rk;cp;ek;C3=r1;c2;ep,r2;c3;ep,...,rk;c1;epasaninterleavedtripleofk-cycles. 3.3CycleModulesWeproduceaT-moduleforeachinterleavedtripleofcycles.WewillneedtosumoverelementsofXwithoneofthethreecoordinatesxed.SincenotriplesotherthanthoseinXareconsideredhere,weshallnotexplicitlywritethiscondition.Weshallplacedotsoverthetwocoordinateswhichvaryinthesummation,e.g.ri;_c;_e,toremindourselvesthatthetriplemustbeanelementofX.Thusalthoughthetwocoordinatesvary,theydonotdosoindependently. Theorem3.3.1 WithNotation3.2.1,xaninterleavedtripleC1,C2,C3ofk-cyclesasinDenition3.2.7. 16

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i For1hkand0j4,thevectorsu1;h:=[[rp;ch;eh]];u2;h:=[[rh;cp;eh]];u3;h:=[[rh;ch+1;ep]];v1;h:=Xrh;_c;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4p[[rh;_c;_e]];v2;h:=X_r;ch;_e2)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(4p[[_r;ch;_e]];v3;h:=X_r;_c;eh2)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(4p[[_r;_c;eh]];[[)]TJ/F28 7.97 Tf 11.825 -1.793 Td[(jp]]spanaT-module.WerefertothisT-moduleasthecyclemoduleofC1,C2,C3anddenoteitWC1;C2;C3. ii TheactionofthegeneratorsEiAjEkofTisasshowninFigures3.2{3.5,wherethesubscriptsaretakenmodulok.InFigures3.2{3.5weabbreviateijk=EiAjEkand ijk=[[)]TJ/F28 7.97 Tf 24.801 -1.793 Td[(i]])]TJ/F27 11.955 Tf 12.175 0 Td[(EiAjEk.TheactionofEiA4EkandEiareomittedastheycanbededucedfrom.1.3and.2.11.Allotheromittedactionsarezero.Proof.InlightofLemma5.3.2,itisenoughtoprovethetheoremforoneofeachtypeofvector.ByLemma2.3.1,Theorem3.1.1,and.2.11,thegeneratorsEiAjEkofTthatactonu1;hinanonzeromannerhaveijk2f011,101,111,23,321,421,431,341,241,441g.Wedonotderiveformulaeforijk2f341;241;441gsincetheiractionisdeducedfrom.1.3,andwedonotdosoforijk=101sinceE1=E1A0E1actsastheidentityonu1;hby2.2.11.ByLemma3.2.3,E2A3E1u1;h=u2;handE3A2E1u1;h=u3;h.TheremainingactionsarededucedusingLemma2.3.4.Since)]TJ/F25 7.97 Tf 298.694 -1.793 Td[(0p=fpgandrp;ch;ehR1p,wendE0A1E1u1;h=[[rp;cp;ep]].Also,E1A1E1u1;h=Xrp;_c;_e6=p;rp;ch;eh[[rp;_c;_e]]=[[)]TJ/F25 7.97 Tf 32.082 -1.793 Td[(1p]])]TJ/F27 11.955 Tf 11.955 0 Td[(u1;h; 17

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E4A2E1u1;h=X_r;ch;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4p[[_r;ch;_e]]=v2;h;E4A3E1u1;h=X_r;_c;eh2)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(4p[[_r;_c;eh]]=v3;h:ByLemma2.3.1,Theorem3.1.1,and.2.11,thegeneratorsEiAjEkofTwhichactonv1;hinanonzeromannerhaveijk2f044,404,124,134,144,214,234,244,314,324,344,414,424,434,444g.Asabove,weneedn'tderiveformulaeforijk2f144;244;344;444;044;404g.ByLemma2.3.4,E3A1E4v1;h=Xrh;_c;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4pE3A1E4[[rh;_c;_e]]=Xrh;_c;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4p[[rh;ch+1;ep]]=Xrh;_c;_e_c6=ch;cpu3;h=n)]TJ/F19 11.955 Tf 11.955 0 Td[(2u3;h;E2A1E4v1;h=n)]TJ/F19 11.955 Tf 11.955 0 Td[(2u2;h;E2A3E4v1;h=Xrh;_c;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4pE2A3E4[[rh;_c;_e]]=Xrh;_c;_e6=rh;cp;eh;rh;ch+1;ep[[rh;_c;_e]]=[[)]TJ/F25 7.97 Tf 30.892 -1.793 Td[(2p]])]TJ/F27 11.955 Tf 11.955 0 Td[(u2;h;E1A3E4v1;h=[[)]TJ/F25 7.97 Tf 30.892 -1.794 Td[(1p]])]TJ/F27 11.955 Tf 11.955 0 Td[(u1;h;E3A2E4v1;h=[[)]TJ/F25 7.97 Tf 30.892 -1.793 Td[(3p]])]TJ/F27 11.955 Tf 11.955 0 Td[(u3;h;E1A2E4v1;h=[[)]TJ/F25 7.97 Tf 30.892 -1.793 Td[(1p]])]TJ/F27 11.955 Tf 11.955 0 Td[(u1;h+1;E4A1E4v1;h=Xrh;_c;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4pE4A1E4[[rh;_c;_e]]=Xrh;_c;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4pXrh;_c0;_e06=rh;_c;_e;rh;ch+1;ep;rh;cp;eh[[rh;_c0;_e0]]=n)]TJ/F19 11.955 Tf 11.955 0 Td[(3v1;h; 18

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E4A2E4v1;h=Xrh;_c;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4pE4A2E4[[rh;_c;_e]]=Xrh;_c;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4pX_r0;_c;_e06=rh;_c;_e;rp;_c;;;_c;ep[[_r0;_c;_e0]]=[[)]TJ/F25 7.97 Tf 30.893 -1.793 Td[(4p]])]TJ/F27 11.955 Tf 11.955 0 Td[(v2;h+1)]TJ/F27 11.955 Tf 11.956 0 Td[(v1;h;E4A3E4v1;h=[[)]TJ/F25 7.97 Tf 30.893 -1.793 Td[(4p]])]TJ/F27 11.955 Tf 11.955 0 Td[(v3;h)]TJ/F27 11.955 Tf 11.955 0 Td[(v1;h:2 [[rp;cp;ep]]2P.................................................................................................................................................................................................011.......................................................................................022.................................................................................................................................................................................................033u3;i)]TJ/F25 7.97 Tf 6.587 0 Td[(1...............................................................................................321...............................................................................................123u1;i...................................................................................................... 111..............................................................................................................231..............................................................................................................132u2;i...................................................................................................... 222......................................................................................................312......................................................................................................213u3;i...................................................................................................... 333...............................................................................................321...............................................................................................123u1;i+1...........................................................................................................................................................................................................................................423..................................................................................................................................................................421..................................................................................................................................................................432.......................................................................................................................................................................................................................412...................................................................................................................................................................................................................431..................................................................................................................................................................413.......................................................................................................................................................................................................................423..................................................................................................................................................................421v2;iv3;iv1;iv2;i+1 Figure3.2:Theactiononu1;i,u2;i,u3;i2WC1;C2;C3 u1;iu2;iu3;iu1;i+1........................................................................................................................................................................................................................................................................................................................................................ 134....................................................................................................................................................................................................................... 234.......................................................................................................................................................................................................................214 n)]TJ/F26 5.978 Tf 5.756 0 Td[(2.............................................................................................................................314 n)]TJ/F26 5.978 Tf 5.756 0 Td[(2............................................................................................................................. 324...................................................................................................................................................................................................... 124v3;iv1;iv2;i+1...................................................................................................... 434)]TJ/F28 7.97 Tf 6.587 0 Td[(I............................................................................................... 424)]TJ/F28 7.97 Tf 6.586 0 Td[(I..................................................................................................................414 n)]TJ/F26 5.978 Tf 5.756 0 Td[(3 Figure3.3:Theactiononv1;i2WC1;C2;C3 u2;i)]TJ/F25 7.97 Tf 6.587 0 Td[(1u3;i)]TJ/F25 7.97 Tf 6.586 0 Td[(1u1;iu2;i........................................................................................................................................................................................................................................................................................................................................ 214....................................................................................................................................................................................................................... 314.......................................................................................................................................................................................................................324 n)]TJ/F26 5.978 Tf 5.756 0 Td[(2.............................................................................................................................124 n)]TJ/F26 5.978 Tf 5.757 0 Td[(2............................................................................................................................. 134...................................................................................................................................................................................................... 234v1;i)]TJ/F25 7.97 Tf 6.586 0 Td[(1v2;iv3;i...................................................................................................... 414)]TJ/F28 7.97 Tf 6.586 0 Td[(I............................................................................................... 434)]TJ/F28 7.97 Tf 6.587 0 Td[(I..................................................................................................................424 n)]TJ/F26 5.978 Tf 5.756 0 Td[(3 Figure3.4:Theactiononv2;i2WC1;C2;C3 19

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u3;i)]TJ/F25 7.97 Tf 6.586 0 Td[(1u1;iu2;iu3;i........................................................................................................................................................................................................................................................................................................................................................ 324....................................................................................................................................................................................................................... 124.......................................................................................................................................................................................................................134 n)]TJ/F26 5.978 Tf 5.756 0 Td[(2.............................................................................................................................234 n)]TJ/F26 5.978 Tf 5.756 0 Td[(2............................................................................................................................. 214...................................................................................................................................................................................................... 314v2;iv3;iv1;i...................................................................................................... 424)]TJ/F28 7.97 Tf 6.587 0 Td[(I............................................................................................... 414)]TJ/F28 7.97 Tf 6.587 0 Td[(I..................................................................................................................434 n)]TJ/F26 5.978 Tf 5.756 0 Td[(3 Figure3.5:Theactiononv3;i2WC1;C2;C3 3.4DecompositionintoIrreducibleModulesInthissectionwedescribethedecompositionofeachcyclemoduleintoirreducibleT-modules. Lemma3.4.1 TheprimarymodulePisanirreducibleT-submoduleofeachcyclemod-ule.Proof.ClearfromTheorems2.3.8and3.3.1.2 Lemma3.4.2 WithNotation3.2.1,xaninterleavedtripleofk-cyclesC1;C2;C3asinDenition3.2.7.Assume1k
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WedenotethisT-moduleW1C1;C2;C3. ii TheactionofthegeneratorsEiAjEkonu11,u12,u13,v11,v12,andv13isasshowninFigures3.6{3.9with=1,wheretheactionofEiA4EkandEiareomittedastheycanbededucedfrom.1.3and.2.11.Allotheromittedactionsarezero. iii Ifn5,thenu11,u12,u13,v11,v12,andv13arelinearlyindependent.Proof.ii:InlightofLemma5.3.2,itsucestoshowthatthegeneratorsEiAjEkofTactonu11andv11asclaimed.AsintheproofofTheorem3.3.1,weneedonlyconsidertheactionofEiAjEkwithijk2f011;111;231;321;421;431gonu11.ByLemma2.3.6andTheorem3.3.1,E0A1E1u11=k[[rp;cp;ep]])]TJ/F27 11.955 Tf 28.184 8.088 Td[(k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[rp;cp;ep]]=0;E1A1E1u11=kXj=1Xrp;_c;_e6=p;rp;cj;ej[[rp;_c;_e]])]TJ/F27 11.955 Tf 28.184 8.088 Td[(k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1E1A1E1[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]]=k[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]])]TJ/F28 7.97 Tf 18.278 14.944 Td[(kXj=1[[rp;cj;ej]])]TJ/F27 11.955 Tf 28.184 8.088 Td[(k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1n)]TJ/F19 11.955 Tf 11.955 0 Td[(2[[)]TJ/F25 7.97 Tf 22.23 -1.793 Td[(1p]]=)]TJ/F27 11.955 Tf 9.299 0 Td[(u11;E2A3E1u11=kXj=1[[rj;cp;ej]])]TJ/F27 11.955 Tf 28.185 8.087 Td[(k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(2p]]=u12;E4A3E1u11=kXj=1X_r;_c;ej2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4p[[_r;_c;ej]])]TJ/F27 11.955 Tf 28.184 8.088 Td[(k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1E4A3E1[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]]=kXj=1X_r;_c;ej2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4p[[_r;_c;ej]])]TJ/F27 11.955 Tf 28.184 8.088 Td[(k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(4p]]=v13;E3A2E1u11=u13;E4A2E1u11=v12;E0A4E4v11=kXj=1Xrj;_c;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4p[[rp;cp;ep]])]TJ/F27 11.955 Tf 28.185 8.088 Td[(k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1E0A4E4[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(4p]]=kn)]TJ/F19 11.955 Tf 11.956 0 Td[(2[[rp;cp;ep]])]TJ/F27 11.955 Tf 28.185 8.087 Td[(k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1n)]TJ/F19 11.955 Tf 11.955 0 Td[(1n)]TJ/F19 11.955 Tf 11.955 0 Td[(2[[rp;cp;ep]]=0; 21

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E4A2E4v11=kXj=10@[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(4p]])]TJ/F47 11.955 Tf 33.005 11.357 Td[(X_r;cj;_e2[[)]TJ/F26 5.978 Tf 8.582 -1.107 Td[(4p]][[_r;cj;_e]]1A)]TJ/F27 11.955 Tf 13.151 8.087 Td[(kn)]TJ/F19 11.955 Tf 11.955 0 Td[(3 n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(4p]]=k[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(4p]])]TJ/F28 7.97 Tf 18.278 14.944 Td[(kXj=1X_r;cj;_e2)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(4p[[_r;cj;_e]])]TJ/F27 11.955 Tf 13.151 8.087 Td[(kn)]TJ/F19 11.955 Tf 11.955 0 Td[(3 n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(4p]]=)]TJ/F27 11.955 Tf 9.298 0 Td[(v11)]TJ/F27 11.955 Tf 11.956 0 Td[(v12;E1A2E4v11=kXj=1[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]])]TJ/F28 7.97 Tf 18.279 14.944 Td[(kXj=1[[rp;cj+1;ej+1]]!)]TJ/F27 11.955 Tf 13.151 8.088 Td[(kn)]TJ/F19 11.955 Tf 11.955 0 Td[(2 n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(1p]]=)]TJ/F28 7.97 Tf 17.614 14.944 Td[(kXj=1[[rp;cj+1;ej+1]]+k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]]=)]TJ/F27 11.955 Tf 9.298 0 Td[(u11:Theotheractionsaresimilarlyveried.i:Byii,W1C1;C2;C3isaT-module.ToshowthatW1C1;C2;C3isirreducible,weshowthatW1C1;C2;C3Tuforanynonzerou2W1C1;C2;C3.FirstsupposeEiu6=0forsomeiwith1i3:Sayi=3.ThenE3u=u132E3W1C1;C2;C3,andE1A2E3u=u11,arenonzeroelementsofE1W1C1;C2;C3.NowW1C1;C2;C3TubyTheorem3.3.1.NowsupposeEiu=0for1i3,sou=1v11+2v12+3v13forsomescalarsii=1;2;3whicharenotallzero.ApplyingE2A1E4,E1A2E4,andE1A3E4touweget.4.6{.4.8.Atleastoneofthesecoecientsisnonzerosince3.4.9hasnononzerosolution.NowW1C1;C2;C3Tu.iii:Wenowshowthatu11,u12,u13,v11,v12,andv13arelinearlyindependentwhenevern5.Wenotethatifn5,thesupportsarenotdistinctsothefollowingargumentwouldn'twork.Supposeu=1u11+2u12+3u13+1v11+2v12+3v13=0.Then1=2=3=0byLemma2.3.5.Nowthefollowing.4.6{3.4.8arezero:E2A1E4u=n)]TJ/F19 11.955 Tf 11.955 0 Td[(21u12)]TJ/F27 11.955 Tf 11.955 0 Td[(2u12)]TJ/F27 11.955 Tf 11.955 0 Td[(3u12=n)]TJ/F19 11.955 Tf 11.955 0 Td[(21)]TJ/F27 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf 11.955 0 Td[(3u12; .4.6 E1A2E4u=n)]TJ/F19 11.955 Tf 11.955 0 Td[(22)]TJ/F27 11.955 Tf 11.955 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(3u11; .4.7 E1A3E4u=n)]TJ/F19 11.955 Tf 11.955 0 Td[(23)]TJ/F27 11.955 Tf 11.955 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(2u11: .4.8 22

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Sinceu11andu12arelinearlyindependent,theircoecientsin.4.6{.4.8arezero.Thusn)]TJ/F19 11.955 Tf 11.955 0 Td[(21)]TJ/F27 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf 11.955 0 Td[(3=n)]TJ/F19 11.955 Tf 11.955 0 Td[(22)]TJ/F27 11.955 Tf 11.955 0 Td[(1)]TJ/F27 11.955 Tf 11.956 0 Td[(3=n)]TJ/F19 11.955 Tf 11.956 0 Td[(23)]TJ/F27 11.955 Tf 11.956 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(2=0:.4.9Equations3.4.9havenonon-zerosolutions.Henceu11,u12,u13,v11,v12,andv13arelinearlyindependent.2Thecasek=n)]TJ/F19 11.955 Tf 11.955 0 Td[(1whichwasexcludedfromLemma3.4.2behavesdierently. Lemma3.4.3 WithreferencetoTheorem3.3.1,supposek=n)]TJ/F19 11.955 Tf 11.955 0 Td[(1.ThenkXi=1uj;i=[[)]TJ/F28 7.97 Tf 30.892 -1.793 Td[(jp]]j=1;2;3;kXi=1vj;i=[[)]TJ/F25 7.97 Tf 30.892 -1.793 Td[(4p]]j=1;2;3:Proof.Clear.2 Lemma3.4.4 WithNotation3.2.1,xaninterleavedtripleofk-cyclesC1;C2;C3asinDenition3.2.7.Assume1
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v3:=kXj=1jv3;j=kXj=1X_r;_c;ej2)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(4pj[[_r;_c;ej]]:WedenotethisT-moduleWC1;C2;C3. ii TheactionofthegeneratorsEiAjEkonthesevectorsisasshowninFigures3.6{3.9,wheretheactionofEiA4EkandEiareomittedastheycanbededucedfrom.1.3and.2.11.Allotheromittedactionsarezero. iii Ifn5,thenu1,u2,u3,v1,v2,andv3arelinearlyindependent.Proof.ii:TheactionfollowsfromTheorem3.3.1.Notethatk>1,sothesumofallkthrootsofunityiszero.Thusforexample,E1A2E4v1=kXj=1jE1A2E4v1;j=kXj=1j[[)]TJ/F25 7.97 Tf 16.378 -1.793 Td[(1p]])]TJ/F27 11.955 Tf 11.955 0 Td[(u1;j+1=kXj=1j[[)]TJ/F25 7.97 Tf 16.377 -1.794 Td[(1p]])]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F25 7.97 Tf 6.587 0 Td[(1kXj=1j+1u1;j+1=)]TJ/F27 11.955 Tf 9.299 0 Td[()]TJ/F25 7.97 Tf 6.586 0 Td[(1u1;E4A3E4v1=kXj=1jE2A3E4v1;j=kXj=1j[[)]TJ/F25 7.97 Tf 16.378 -1.793 Td[(4p]])]TJ/F27 11.955 Tf 11.956 0 Td[(v3;j)]TJ/F27 11.955 Tf 11.955 0 Td[(v1;j=)]TJ/F27 11.955 Tf 9.298 0 Td[(v3)]TJ/F27 11.955 Tf 11.955 0 Td[(v1E4A2E1u1=kXj=1jE4A2E1u1;j=kXj=1jv2;j=v2:Theremainingactionsarecomputedsimilarly.i:ArguingasinLemma3.4.4givesthatspanu1;u2;u3v1;v2;v3isclosedundertheactionofthegeneratorsEiAjEkofTandthatWC1;C2;C3isirreducible. 24

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iii:Toshowlinearlyindependence,letv=1u1+2u2+3u3+1v1+2v2+3v3=0.ByLemma2.3.5wehave1=2=3=0.Toshowthat1=2=3=0,applyE1A2E4,E1A3E4,andE2A1E4tovtond)]TJ/F27 11.955 Tf 9.299 0 Td[(1+n)]TJ/F19 11.955 Tf 11.955 0 Td[(22)]TJ/F27 11.955 Tf 11.955 0 Td[(3u1=0;)]TJ/F27 11.955 Tf 9.299 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(2+n)]TJ/F19 11.955 Tf 11.955 0 Td[(23u1=0;n)]TJ/F19 11.955 Tf 11.956 0 Td[(21)]TJ/F27 11.955 Tf 11.956 0 Td[(2)]TJ/F27 11.955 Tf 11.955 0 Td[(3u2=0:Thus)]TJ/F27 11.955 Tf 9.299 0 Td[(1+2n)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf 11.955 0 Td[(3=)]TJ/F27 11.955 Tf 9.298 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(2+n)]TJ/F19 11.955 Tf 11.955 0 Td[(23=n)]TJ/F19 11.955 Tf 11.955 0 Td[(21)]TJ/F27 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf 11.955 0 Td[(3=0:Solvingthesethreeequationsgives1=2=3=0.Henceu1,u2,u3,v1,v2,andv3arelinearlyindependent.2 25

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u1.............................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(111.............................................................................................................................................................................................................................................231.............................................................................................................................................................................................................................................132u2.............................................................................................)]TJ/F25 7.97 Tf 6.587 0 Td[(222......................................................................................................................................................................................................................................312......................................................................................................................................................................................................................................213u3.............................................................................................)]TJ/F25 7.97 Tf 6.587 0 Td[(333..................................................................................................................................................................421..................................................................................................................................................................432..................................................................................................................................................................413......................................................................................................................................................................................................................................................................................................431..........................................................................................................................................................................................................................................................................................412................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................423v2v3v1...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................123.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................)]TJ/F26 5.978 Tf 5.756 0 Td[(1321 Figure3.6:Theactiononui2WC1;C2;C3 u3u1u2................................................................................................................................................................................................................................................................................................................314 n)]TJ/F26 5.978 Tf 5.757 0 Td[(2................................................................................................................................................................................................................................................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(324...............................................................................................)]TJ/F28 7.97 Tf 6.587 0 Td[(124...............................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(134................................................................................................................................................................................................................................................................................................................)]TJ/F25 7.97 Tf 6.587 0 Td[(234................................................................................................................................................................................................................................................................................................................214 n)]TJ/F26 5.978 Tf 5.756 0 Td[(2v1.............................................................................................................................................................................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(424)]TJ/F28 7.97 Tf 6.586 0 Td[(Iv2.....................................................................................................................................................................................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(434)]TJ/F28 7.97 Tf 6.587 0 Td[(Iv3.............................................................................................414 n)]TJ/F26 5.978 Tf 5.756 0 Td[(3 Figure3.7:Theactiononv12WC1;C2;C3 u1u2u3................................................................................................................................................................................................................................................................................................................124 n)]TJ/F26 5.978 Tf 5.757 0 Td[(2................................................................................................................................................................................................................................................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(134...............................................................................................)]TJ/F28 7.97 Tf 6.587 0 Td[()]TJ/F26 5.978 Tf 5.756 0 Td[(1214...............................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(234................................................................................................................................................................................................................................................................................................................)]TJ/F28 7.97 Tf 6.586 0 Td[()]TJ/F26 5.978 Tf 5.756 0 Td[(1314................................................................................................................................................................................................................................................................................................................324 n)]TJ/F26 5.978 Tf 5.756 0 Td[(2v2.............................................................................................................................................................................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(434)]TJ/F28 7.97 Tf 6.586 0 Td[(Iv3.....................................................................................................................................................................................................................................................)]TJ/F26 5.978 Tf 5.756 0 Td[(1)]TJ/F25 7.97 Tf 6.587 0 Td[(414)]TJ/F28 7.97 Tf 6.587 0 Td[(Iv1.............................................................................................424 n)]TJ/F26 5.978 Tf 5.756 0 Td[(3 Figure3.8:Theactiononv22WC1;C2;C3 u2u3u1................................................................................................................................................................................................................................................................................................................234 n)]TJ/F26 5.978 Tf 5.757 0 Td[(2................................................................................................................................................................................................................................................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(214...............................................................................................)]TJ/F25 7.97 Tf 6.587 0 Td[(314...............................................................................................)]TJ/F28 7.97 Tf 6.587 0 Td[()]TJ/F26 5.978 Tf 5.756 0 Td[(1324................................................................................................................................................................................................................................................................................................................)]TJ/F25 7.97 Tf 6.587 0 Td[(124................................................................................................................................................................................................................................................................................................................134 n)]TJ/F26 5.978 Tf 5.756 0 Td[(2v3.............................................................................................................................................................................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(414)]TJ/F28 7.97 Tf 6.586 0 Td[(Iv1.....................................................................................................................................................................................................................................................)]TJ/F25 7.97 Tf 6.586 0 Td[(424)]TJ/F28 7.97 Tf 6.587 0 Td[(Iv2.............................................................................................434 n)]TJ/F26 5.978 Tf 5.756 0 Td[(3 Figure3.9:Theactiononv32WC1;C2;C3 26

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Lemma3.4.5 WithNotation3.2.1,xaninterleavedtripleofk-cyclesC1;C2;C3asinDenition3.2.7.Letandbedistinctkthrootsofunity.ThenWC1;C2;C3andWC1;C2;C3arenon-isomorphicT-modules.Moreover,WC1;C2;C3andWC1;C2;C3areorthogonal.Proof.Suppose6=1.ObservethatE1A2E3A1E2A3E1u1=E1A2E3A1E2A3E1kXj=1j[[rp;cj;ej]]=kXj=1j[[rp;cj+1;ej+1]]=)]TJ/F25 7.97 Tf 6.586 0 Td[(1kXj=1j+1[[rp;cj+;ej+1]]=)]TJ/F25 7.97 Tf 6.586 0 Td[(1u1:AlsoE1A2E3A1E2A3E1actsaszeroonu2,u3,v1,v2,v3.Similarly,if6=1,thenE1A2E3A1E2A3E1u1=)]TJ/F25 7.97 Tf 6.586 0 Td[(1u1,andE1A2E3A1E2A3E1actsaszeroonu2,u3,v1,v2,v3.Thustheresultholdsinthiscase.Suppose=1.NowbyLemma2.3.6andTheorem3.1.1,E1A2E3A1E2A3E1u11=E1A2E3A1E2A3E1kXj=1[[rp;cj;ej]])]TJ/F27 11.955 Tf 23.632 8.088 Td[(k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(1p]]=kXj=1[[rp;cj+1;ej+1]])]TJ/F27 11.955 Tf 23.632 8.088 Td[(k n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]]=u11:AlsoE1A2E3A1E2A3E1actsaszeroonu12,u13,v11,v12,v13.ItfollowsthatWC1;C2;C3andWC1;C2;C3arenon-isomorphic.TheorthogonalityofthesetwomodulesfollowsfromLemma2.3.3.2 Theorem3.4.6 WithNotation3.2.1,xaninterleavedtripleofk-cyclesC1;C2;C3. i Ifk6=n)]TJ/F19 11.955 Tf 11.392 0 Td[(1,thenWC1;C2;C3hasorthogonaldirectdecompositionintoirreducible 27

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T-modulesWC1;C2;C3=PM2Ck=1WC1;C2;C3: ii Ifk=n)]TJ/F19 11.955 Tf 11.392 0 Td[(1,thenWC1;C2;C3hasorthogonaldirectdecompositionintoirreducibleT-modulesWC1;C2;C3=PM2Ck=1;6=1WC1;C2;C3:Proof.ItisclearfromLemmas2.3.3and3.4.5thatthesumsiniandiiareorthogonalandhencedirect.Firstsupposek
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complexmthrootof1.Fori=1;:::;`setu;`1;i=mXj=1j[[rp;cj`+i;ej`+i]];u;`2;i=mXj=1j[[rj`+i;cp;ej`+i]];u;`3;i=mXj=1j[[rj`+i)]TJ/F25 7.97 Tf 6.587 0 Td[(1;cj`+i;ep]];v;`1;i=mXj=1Xrj`+i;_c;_e2)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(4pj[[rj`+i;_c;_e]];v;`2;i=mXj=1X_r;cj`+i;_e2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4pj[[_r;cj`+i;_e]];v;`3;i=mXj=1X_r;_c;ej`+i2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4pj[[_r;_c;ej`+i]]:Fort=1;2;3,letUt=fu;`t;ig`i=1andVt=fv;`t;ig`i=1.Then[3t=1Ut[[3t=1Vt[f[[)]TJ/F28 7.97 Tf 11.825 -1.794 Td[(t]]g4t=0spansaT-module.Moreoverifn5,thenthefollowinghold. i Ifk
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PartsiandiiofLemma3.5.1failifn5sincethedimensionofE4Vistoosmall.WenotethatthecyclemoduleWC1;C2;C3appearsinLemma3.5.1inthecasem=1,`=kinwhichcase=1.TheirreduciblesubmoduleWC1;C2;C3appearsinLemma3.5.1inthecasem=k,`=1. 3.6CollectingCycleModulesWebeginbyextendingNotation3.2.1.Toavoiddegeneratesituations,onlyconsiderLatinsquareswithorderatleast5. Notation3.6.1 LetLdenoteaLatinsquareofordern5andwithsymbolsetf1;2;:::;ng.LetXdenotethesetfi;j;Li;jj1i;jng.LetMdenotetheBose-MesneralgebraofL.Fixp=rp;cp;ep2X,andletTdenotethesubconstitutentalgebraofMwithrespecttorp;cp;ep.LetI1,I2,...,ImdenotetheinterleavedcyclesofLwithrespecttop.DenotetheelementsofIjasCj1,Cj2,Cj3.UseXjtorefertoobjectXassociatedwithWCj1;Cj2;Cj3;forexampleu1j. Lemma3.6.2 WithNotation3.6.1.FixtwodistinctinterleavedtriplesofcyclesC1,C2,C3andC01,C02,C03.SupposetheCiarek-cyclesandtheC0iarek0-cycles.Let`beapositiveintegersuchthat`jkand`jk0,andletbean`throotofunity.ThenWC1;C2;C3andWC01;C02;C03areisomorphicT-modules.Proof.Notethatneitherknork0isn)]TJ/F19 11.955 Tf 10.277 0 Td[(1sincetherearetwodistinctinterleavedtriplesofcycles.ThusW1C1;C2;C3andW1C01;C02;C03arebothdened.However,thecase=1neednotbetreatedseparately.SupposethemodulesWC1;C2;C3andWC01;C02;C03haverespectivebasesfu1;u2;u3;v1;v2;v3gandfs1;s2;s3;t1;t2;t3g.Denealinearmap:WC1;C2;C3!WC01;C02;C03byui=siandvi=tii3.Itisclearthatisabijection.ToshowthatforallA2Tandv2V,Av=AvitisenoughtotreatthecasewhereAisoftheformEiAjEkandvisoneofthebasiselementsuiorvi.Forexample,E3A2E1u1=v2=u3=v3=E3A2E1s1=E3A2E1u1: 30

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TheresultfollowsfromTheorem3.3.1Figures3.6{3.9.2 Lemma3.6.3 WithNotation3.6.1.Ifandarebothprimitive`throotsofunity,thenmultWC1;C2;C3=multWC1;C2;C3:Proof.Clearsincethemultiplicityofeachisthenumberofinterleavedcycleswithlengthdivisibleby`.2Incontrasttothesituationfornon-isomorphicirreducibleT-modules,theisomorphicirreducibleT-modulescontainedindistinctcyclemodulesaregenerallynotorthogonaltooneanother.Itturnsoutthatwiththeexceptionofthedependenciesnotedinthefollowinglemma,isomorphicirreducibleT-modulesconstructedsofararelinearlyinde-pendent. Lemma3.6.4 WithNotation3.6.1,supposem1fori=1,2,3,mXj=1u1ij=0andmXj=1v1ij=0:Proof.SupposethatCj1haslengthkjhm.Recallthatu11j=Xrp;_c;_e2Cj1[[rp;_c;_e]])]TJ/F27 11.955 Tf 20.874 8.088 Td[(kj n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(1p]];somXj=1u11j=mXj=1Xrp;_c;_e2Cj1[[rp;_c;_e]])]TJ/F28 7.97 Tf 16.844 14.944 Td[(mXj=1kj n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]]=[[)]TJ/F25 7.97 Tf 30.892 -1.793 Td[(1p]])]TJ/F19 11.955 Tf 11.955 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]]=0:Similarly,Pmj=1u12j=Pmj=1u13j=0.Notethatv11j=kjXh=1Xrhj;_c;_e2[[)]TJ/F26 5.978 Tf 8.582 -1.107 Td[(4p]][[rhj;_e;_c]])]TJ/F27 11.955 Tf 20.874 8.088 Td[(kj n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(4p]]: 31

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ThusmXj=1v11j=mXj=1kjXh=1Xrhj;_c;_e2[[)]TJ/F26 5.978 Tf 8.582 -1.107 Td[(4p]][[rhj;_c;_e]])]TJ/F19 11.955 Tf 11.955 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(4p]]=[[)]TJ/F25 7.97 Tf 30.892 -1.793 Td[(4p]])]TJ/F19 11.955 Tf 11.955 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(4p]]=0:Similarly,Pmj=1v12j=Pmj=1v13j=0.2Whenm=1,Lemma3.6.4restatesLemma3.4.3. Lemma3.6.5 WithNotation3.6.1,thefollowingsetislinearlyindependent:[m)]TJ/F25 7.97 Tf 6.586 0 Td[(1j=1fu11j;u12j;u13j;v11j;v12j;v13jg:Proof.For1jm,letw11j=u11j+kj n)]TJ/F19 11.955 Tf 11.955 0 Td[(1[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]]=kjXh=1[[rp;chj;ehj]]:Sincetherstcoordinateofthesupportofeachw11jaredisjoint,thesetfw11jgmj=1islinearlyindependentbyLemma2.3.5.Itfollowsthatfu11jgm)]TJ/F25 7.97 Tf 6.587 0 Td[(1j=1islinearlyindependent.Similarlyfu12jgm)]TJ/F25 7.97 Tf 6.587 0 Td[(1j=1andfu13jgm)]TJ/F25 7.97 Tf 6.587 0 Td[(1j=1arelinearlyindependentsets.Letu=3Xh=1m)]TJ/F25 7.97 Tf 6.587 0 Td[(1Xj=1jhu1hj+3Xh=1m)]TJ/F25 7.97 Tf 6.587 0 Td[(1Xj=1jhv1hj; 32

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andsupposeu=0.Thenj1=j2=j3=0byLemma2.3.5andtheabove.ApplyingE1A2E4,E1A3E4,andE2A1E4tougives0=)]TJ/F27 11.955 Tf 9.298 0 Td[(1+n)]TJ/F19 11.955 Tf 11.955 0 Td[(22)]TJ/F27 11.955 Tf 11.955 0 Td[(3u11++)]TJ/F27 11.955 Tf 9.299 0 Td[(m)]TJ/F25 7.97 Tf 6.586 0 Td[(11+n)]TJ/F19 11.955 Tf 11.956 0 Td[(2m)]TJ/F25 7.97 Tf 6.586 0 Td[(12)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.586 0 Td[(13u11m)]TJ/F19 11.955 Tf 11.955 0 Td[(1;0=)]TJ/F27 11.955 Tf 9.298 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(2+n)]TJ/F19 11.955 Tf 11.955 0 Td[(23u11++)]TJ/F27 11.955 Tf 9.299 0 Td[(m)]TJ/F25 7.97 Tf 6.586 0 Td[(11)]TJ/F27 11.955 Tf 11.956 0 Td[(m)]TJ/F25 7.97 Tf 6.586 0 Td[(12+n)]TJ/F19 11.955 Tf 11.955 0 Td[(2m)]TJ/F25 7.97 Tf 6.586 0 Td[(13u11m)]TJ/F19 11.955 Tf 11.955 0 Td[(1;0=n)]TJ/F19 11.955 Tf 11.955 0 Td[(21)]TJ/F27 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf 11.955 0 Td[(3u12++n)]TJ/F19 11.955 Tf 11.955 0 Td[(2m)]TJ/F25 7.97 Tf 6.587 0 Td[(11)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(12)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(13u12m)]TJ/F19 11.955 Tf 11.955 0 Td[(1:Hencefor1jm,)]TJ/F27 11.955 Tf 9.299 0 Td[(j1+n)]TJ/F19 11.955 Tf 11.955 0 Td[(2j2)]TJ/F27 11.955 Tf 11.955 0 Td[(j3=0;)]TJ/F27 11.955 Tf 9.299 0 Td[(j1)]TJ/F27 11.955 Tf 11.955 0 Td[(j2+n)]TJ/F19 11.955 Tf 11.955 0 Td[(2j3=0;n)]TJ/F19 11.955 Tf 11.956 0 Td[(2j1)]TJ/F27 11.955 Tf 11.956 0 Td[(j2)]TJ/F27 11.955 Tf 11.955 0 Td[(j3=0:AsintheproofofLemma3.4.2,itfollowsthatj1=j2=j3=0foralljjm)]TJ/F19 11.955 Tf 11.955 0 Td[(1.2ThechoiceofomissioninLemma3.6.5iswasarbitrary. Lemma3.6.6 WithNotation3.6.1,letbeaprimitive`throotofunityotherthan1.LetIi1,Ii2,...,Iipbeallinterleavedcycleswithorderdivisibleby`.Thenthefollowingsetislinearlyindependent:[pj=1fu1ij;u2ij;u3ij;v1ij;v2ij;v3ijg:Proof.Letu=3Xh=1pXj=1jhuhij+3Xh=1pXj=1jhvhij; 33

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andsupposeu=0.Thenj1=j2=j3=0jpbyLemma2.3.5.ApplyingE1A2E4,E1A3E4,andE2A1E4tougives0=)]TJ/F27 11.955 Tf 9.299 0 Td[(1+n)]TJ/F19 11.955 Tf 11.955 0 Td[(22)]TJ/F27 11.955 Tf 11.955 0 Td[(3u1i1++)]TJ/F27 11.955 Tf 9.299 0 Td[(p1+n)]TJ/F19 11.955 Tf 11.955 0 Td[(2p2)]TJ/F27 11.955 Tf 11.955 0 Td[(p3u1ip;0=)]TJ/F27 11.955 Tf 9.299 0 Td[(1)]TJ/F27 11.955 Tf 11.956 0 Td[(2+n)]TJ/F19 11.955 Tf 11.955 0 Td[(23u1i1++)]TJ/F27 11.955 Tf 9.299 0 Td[(p1)]TJ/F27 11.955 Tf 11.955 0 Td[(p2+n)]TJ/F19 11.955 Tf 11.956 0 Td[(2p3u1ip;0=n)]TJ/F19 11.955 Tf 11.955 0 Td[(21)]TJ/F27 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf 11.955 0 Td[(3u2i1++n)]TJ/F19 11.955 Tf 11.955 0 Td[(2p1)]TJ/F27 11.955 Tf 11.955 0 Td[(p2)]TJ/F27 11.955 Tf 11.956 0 Td[(p3u2ip:Butthesetfuhijgpj=1islinearlyindependentbyLemma2.3.5.Hence,)]TJ/F27 11.955 Tf 9.299 0 Td[(j1+n)]TJ/F19 11.955 Tf 11.955 0 Td[(2j2)]TJ/F27 11.955 Tf 11.955 0 Td[(3j=0;)]TJ/F27 11.955 Tf 9.299 0 Td[(j1)]TJ/F27 11.955 Tf 11.956 0 Td[(j2+n)]TJ/F19 11.955 Tf 11.955 0 Td[(2j3=0;n)]TJ/F19 11.955 Tf 11.955 0 Td[(2j1)]TJ/F27 11.955 Tf 11.955 0 Td[(j2)]TJ/F27 11.955 Tf 11.955 0 Td[(j3=0:Thus1j=2j=3j=0foralljjp,asintheproofofLemma3.4.4.2TheuijarealwayspairwiseorthogonalandorthogonaltoallvijbyLemma2.3.5.OnemayapplyGram-Schmidtorthonormalizationproceduretothebasisoftheabovelemmaiforthogonalityisrequired.Theresultingbasiswillnolongerbesonicelyrelatedtothecyclemodules. Corollary3.6.7 WithNotation3.6.1,dimmXj=1WCj1;Cj2;Cj3=6n)]TJ/F19 11.955 Tf 11.956 0 Td[(7: 34

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Proof.RecallthatWCj1;Cj2;Cj3P?=8>>>>>><>>>>>>:Mkj=1WCj1;Cj2;Cj3ifkj6=n)]TJ/F19 11.955 Tf 11.955 0 Td[(1,Mkj=16=1WCj1;Cj2;Cj3ifkj=n)]TJ/F19 11.955 Tf 11.955 0 Td[(1.Hence,dimWCj1;Cj2;Cj3P?=8><>:6kjifkj6=n)]TJ/F19 11.955 Tf 11.956 0 Td[(1,6kj)]TJ/F19 11.955 Tf 11.955 0 Td[(6ifkj=n)]TJ/F19 11.955 Tf 11.956 0 Td[(1.WiththeconventionthatWCj1;Cj2;Cj3=0ifisnotakjthrootofunity,mXj=1WCj1;Cj2;Cj3P?=8>>>>>><>>>>>>:MmXj=1WCj1;Cj2;Cj3ifkj6=n)]TJ/F19 11.955 Tf 11.955 0 Td[(1,Mkj=16=1WCj1;Cj2;Cj3ifkj=n)]TJ/F19 11.955 Tf 11.955 0 Td[(1.NowbyLemma3.6.4,dimPmj=1WCj1;Cj2;Cj3P?Pmj=16kj)]TJ/F19 11.955 Tf 11.956 0 Td[(6,andbyLemmas3.6.5and3.6.6,dimPmj=1WCj1;Cj2;Cj3P?Pmj=16kj)]TJ/F19 11.955 Tf 11.955 0 Td[(6.Thus,dimmXj=1WCj1;Cj2;Cj3P?=8><>:Pmj=16kj)]TJ/F19 11.955 Tf 11.955 0 Td[(6ifkj6=n)]TJ/F19 11.955 Tf 11.955 0 Td[(1,6n)]TJ/F19 11.955 Tf 11.955 0 Td[(1)]TJ/F19 11.955 Tf 11.955 0 Td[(6ifkj=n)]TJ/F19 11.955 Tf 11.955 0 Td[(1.SincePmj=1kj=n)]TJ/F19 11.955 Tf 11.955 0 Td[(1,wendinbothcasesthatdimmXj=1WCj1;Cj2;Cj3P?=6n)]TJ/F19 11.955 Tf 11.955 0 Td[(12:Adding5,thedimensionoftheprimarymoduleP,givestheresult.2 35

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Corollary3.6.8 WithNotation3.6.1,dimEimXj=1WCj1;Cj2;Cj3=8>>>><>>>>:1ifi=0,n)]TJ/F19 11.955 Tf 11.955 0 Td[(1ifi=1;2;3,3n)]TJ/F19 11.955 Tf 11.955 0 Td[(5ifi=4.Proof.Notethat[[p]]isabasisforE0Pmj=1WCj1;Cj2;Cj3.Fori=1,thesetofvectors,f[[rp;_c;_e]]jrp;_c;_e2XgformsabasisforE1Pmj=1WCj1;Cj2;Cj3ofdimensionn)]TJ/F19 11.955 Tf 12.075 0 Td[(1.Similarlyfori=2;3.Fori=4,notethatPmj=1WCj1;Cj2;Cj3istheorthogonalsummXj=1WCj1;Cj2;Cj3=4Mi=0EimXj=1WCj1;Cj2;Cj3:HencethedimensionofE4Pmj=1WCj1;Cj2;Cj3=6n)]TJ/F19 11.955 Tf 13.092 0 Td[(7)]TJ/F19 11.955 Tf 13.093 0 Td[(3n)]TJ/F19 11.955 Tf 13.092 0 Td[(1)]TJ/F19 11.955 Tf 13.092 0 Td[(1=3n)]TJ/F19 11.955 Tf 13.092 0 Td[(5.2 Corollary3.6.9 WithNotation3.6.1,fori=0,1,2,3,EiVmXj=1WCj1;Cj2;Cj3:Proof.ByCorollary3.6.8,E0V[[p]].Eachv2)]TJ/F28 7.97 Tf 7.314 -1.793 Td[(ii=1;2;3isinsomecycle,so[[v]]isinthecorrespondingcyclemodule.HenceEiVmXj=1WCj1;Cj2;Cj3.2 3.7TheFourthSubconstituentThecyclemodulesdonotaccountforallirreducibleT-modules,asE4Visnotcontainedintheirsum.InthissectionweshowthatthepartofE4Vnotcontainedinthecyclemodulesdecomposesintomutuallyisomorphicone-dimensionalT-modules. Lemma3.7.1 WithNotation3.6.1,pickanynonzerovectorv2E4Vwhichisorthogo-naltoallWCj1;Cj2;Cj3jm.Writev=Px2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4px[[x]].Thenfor1i3and 36

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for1kn,k6=pi,Xx2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4pxi=kx=0.Proof.For0i3,EiVPm`=1WC`1;C`2;C`3,andEiAjE4v2EiV.ThusEiAjE4v=0i;j3.Hence,0=E1A2E4v=Xx2)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(4px[[rp;x;Lrp;x]]=Xk6=cpXx=kx[[rp;x;Lrp;x]]:Fork16=k2,thevectorsXx=k1[[rp;x;Lrp;x]]andXx=k2[[rp;x;Lrp;x]]arelinearlyindependentbyLemma2.3.5.ThusXx=kx[[rp;x;Lrp;x]]=0,henceXxj=kx=0forj=2.SimilarlyXxj=kx=0forj=1;3.2 Theorem3.7.2 WithNotation3.6.1,pickanynonzerovectorv2E4Vwhichisor-thogonaltoallWCj1;Cj2;Cj3jm.Thenvspansaone-dimensionalirreducibleT-module.WedenotethisT-modulebyFv.TheactionofthegeneratorsEiAjEkonthiselementisE4A0E4v=v,E4AiE4v=)]TJ/F27 11.955 Tf 9.298 0 Td[(vfori=1;2,and3,E4A4E4v=2v,EiAjEkv=0forallotheri;j,andk.Proof.ThegeneratorsEiAjEkofTthatactonvinanonzeromannerhaveijk2f044;124;134;234;214;314;324;404;414;424;434;444g.Bythechoiceofv,EiAjE4v=0 37

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for1i;j3,i6=j.By.1.1,E4A0E4v=E4v=v.NowE4A1E4v=Xx2)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(4pxE4A1E4[[x]]=Xx2)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(4pxXy2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4px[[y]])]TJ/F19 11.955 Tf 11.955 0 Td[([[x]]=Xx2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4py1=x1x[[y]])]TJ/F47 11.955 Tf 18.672 11.357 Td[(Xx2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4px[[x]]=)]TJ/F47 11.955 Tf 16.015 11.357 Td[(Xx2)]TJ/F26 5.978 Tf 5.288 -1.107 Td[(4px[[x]]=)]TJ/F27 11.955 Tf 9.298 0 Td[(v:SinceXx2)]TJ/F26 5.978 Tf 5.289 -1.107 Td[(4py1=x1x[[y]]=0byLemma3.7.1.Similarly,E4A2E4v=E4A3E4v=)]TJ/F27 11.955 Tf 9.298 0 Td[(v:Also,E4A4E4v=E4J)]TJ/F27 11.955 Tf 11.955 0 Td[(A0)]TJ/F27 11.955 Tf 11.956 0 Td[(A1)]TJ/F27 11.955 Tf 11.955 0 Td[(A2)]TJ/F27 11.955 Tf 11.956 0 Td[(A3E4v=E4JE4)]TJ/F27 11.955 Tf 11.956 0 Td[(v+3v=2v:NotethatE4JE4v=0sincev?[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(4p]].2 Corollary3.7.3 WithreferencetoTheorem3.7.2,theT-moduleisomorphismclassofFvisindependentofthechoiceofv.ThemultiplicityofFvisn2)]TJ/F19 11.955 Tf 11.955 0 Td[(6n+7.Proof.ThatallFvareisomorphicfollowsfromTheorem3.7.2byanargumentsimilartothatintheproofofLemma3.6.2.WenowconsiderthemultiplicityofFv.NotethatE4V=E4mXj=1WCj1;Cj2;Cj3ME4mXj=1WCj1;Cj2;Cj3?:ThedimensionofE4Visn2)]TJ/F19 11.955 Tf 13.054 0 Td[(3n+2,andthedimensionofE4mXj=1WCj1;Cj2;Cj3is3n)]TJ/F19 11.955 Tf 12.4 0 Td[(5.Hencethethedimensionofthecomplementisn2)]TJ/F19 11.955 Tf 12.4 0 Td[(6n+7.ByLemma3.7.1,theorthogonalcomplementisthesumofisomorphiccopiesofone-dimensionalmodules 38

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isomorphictoFv.ThusFvhasmultiplicityn2)]TJ/F19 11.955 Tf 11.955 0 Td[(6n+7.2 3.8OtherResultsWepresentafewadditionalresults.Werstnotethatwenowhaveacompletedecom-positionofVintoirreducibleT-modules. Lemma3.8.1 WithNotation3.6.1,letv1;v2;:::;vn2)]TJ/F25 7.97 Tf 6.587 0 Td[(6n+7beanorthogonalbasisforPmj=1WCj1;Cj2;Cj3?.ThenV=PMmXj=1Xkj=1WCj1;Cj2;Cj3n2)]TJ/F25 7.97 Tf 6.586 0 Td[(6n+7Mi=1Fvi:Proof.StraightforwardfromTheorem3.4.6,Lemma3.6.6,andCorollaries3.6.7,3.6.8,3.6.9,3.7.3.2 Lemma3.8.2 Theisomorphismclassofeachnon-primaryirreducibleT-moduleWisuniquelydeterminedbytheeigenvalueofE1A2E3A1E2A2E1associatedwithE1W.Proof.ClearfromLemma3.6.2andCorollary3.7.3.2 3.9CayleyTablesofFiniteGroupsTheCayleytableofanitegroupisaLatinsquare.Wedescribethecyclestructureoftheseexamples. Theorem3.9.1 LetGbeanitegroup,andletLdenotetheCayleytableofG.Considerthesubconstituentalgebrawithrespecttog;h;ghoftheBose-MesneralgebraofL.ThenthecyclestructureofLwithrespecttog;h;ghis12jGj)]TJ/F28 7.97 Tf 8.939 0 Td[()]TJ/F25 7.97 Tf 6.587 0 Td[(1=2,whereisthenumberofelementsinGoforder2. 39

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Proof.ComputeE1A3E2A1E3A2E1g;b;gb=E1A3E2A1E3ghb)]TJ/F25 7.97 Tf 6.587 0 Td[(1;b;gh=E1A3E2ghb)]TJ/F25 7.97 Tf 6.587 0 Td[(1;h;ghb)]TJ/F25 7.97 Tf 6.586 0 Td[(1h=g;hb)]TJ/F25 7.97 Tf 6.586 0 Td[(1h;ghb)]TJ/F25 7.97 Tf 6.587 0 Td[(1h:Nowrepeatthiscomputation:E1A3E2A1E3A2E1g;hb)]TJ/F25 7.97 Tf 6.587 0 Td[(1h;ghb)]TJ/F25 7.97 Tf 6.586 0 Td[(1h=E1A3E2A1E3gbh)]TJ/F25 7.97 Tf 6.587 0 Td[(1;hb)]TJ/F25 7.97 Tf 6.586 0 Td[(1h;gh=E1A3E2gbh)]TJ/F25 7.97 Tf 6.587 0 Td[(1;h;gb=g;b;gb:ThustheorderofE1A3E2A1E3A2E1isoneortwo.Observethatg;b;gbformsaone-cycleifandonlyifb=hb)]TJ/F25 7.97 Tf 6.586 0 Td[(1hifandonlyifhb2=e.SincebcanbeanyelementofGotherthanh,thereisexactlyoneone-cycleforeachelementofGoforder2sincewemayexcludetheidentity,whichhasorder1.2 Corollary3.9.2 WithreferencetoTheorem3.9.1,thecyclestructureis1jGj)]TJ/F25 7.97 Tf 8.939 0 Td[(1ifandonlyifG=Z2Z2Z2.Proof.Thecyclestructureis1jGj)]TJ/F25 7.97 Tf 8.939 0 Td[(1ifandonlyifb=hb)]TJ/F25 7.97 Tf 6.586 0 Td[(1hforallbifandonlyifhb)]TJ/F25 7.97 Tf 6.587 0 Td[(12=eforallb2Gnhifandonlyifx2=eforallx2GneifandonlyifG=Z2Z2Z2.2 3.10SmallLatinSquaresWereportresultsforLatinsquaresofordern4whichwereexcludedfrommuchofourdiscussionuptothispoint.WeshallseethatthesmallsizeoftheseLatinsquaresforcestheirreducibleT-modulestohavesmallerdimensionthatpredictedforlargerLatinsquares. Example3.10.1 ThereisauniqueLatinsquareoforder1,namely.Notethat 40

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A1=A2=A3=A4=0.Thestandardmodulehasbasis[;1;1]=[)]TJ/F25 7.97 Tf 39.969 -1.793 Td[(0;1;1],soitisspannedbytheprimarymodule. Example3.10.2 TheuniqueLatinsquareoforder2uptomainclassequivalenceistheCayleytableofZ2:0@12211A:NotethatA1=A2andA4=0.Withrespecttoanybasepointp,thereisasingleinterleaved1-cycleC1;C2;C3.Thestandardmoduleisthesummoftheprimarymodulewithbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(0p]];[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]];[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(3p]]g,andW1C1;C2;C3whichhasbasisfu11g. Example3.10.3 TheuniqueLatinsquareoforder3uptomainclassequivalenceistheCayleytableofZ3:0BBB@1232313121CCCA:Withrespecttoanybasepointp,thereisasingleinterleaved2-cycleC1;C2;C3andtwoelements)]TJ/F25 7.97 Tf 55.992 -1.793 Td[(4p.ThestandardmoduleisthesumoftheprimarymodulePwithbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(0p]];[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(1p]];[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(2p]];[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(3p]];[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(4p]]g,W)]TJ/F25 7.97 Tf 6.587 0 Td[(1C1;C2;C3withbasisfu)]TJ/F25 7.97 Tf 6.587 0 Td[(11;u)]TJ/F25 7.97 Tf 6.586 0 Td[(12;u)]TJ/F25 7.97 Tf 6.587 0 Td[(13;v3)]TJ/F25 7.97 Tf 6.586 0 Td[(1g.Fori=1,2,3,[[)]TJ/F28 7.97 Tf 66.529 -1.793 Td[(ip]]isthesumoftwovectorsandu)]TJ/F25 7.97 Tf 6.587 0 Td[(1iisthedierenceofthesametwovectors.Similarlyforv3)]TJ/F25 7.97 Tf 6.586 0 Td[(1and[[)]TJ/F25 7.97 Tf 34.586 -1.794 Td[(4p]]. Example3.10.4 TheCayleytableofZ4representsoneofthetwomainclassofLatinsquaresoforder4:0BBBBB@12342341341241231CCCCCA:Withrespecttoanybasepointp,thereisaaninterleaved1-cycleC11;C12;C13andandinterleaved2-cycleC21;C22;C23.Thestandardmoduleisthesumoftheprimarymod-ulePwithbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(0p]];[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(1p]];[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(2p]];[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(3p]];[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(4p]]g,W1C21;C22;C23withbasisfu11;u12;u13;v11;v12g,andW)]TJ/F25 7.97 Tf 6.586 0 Td[(1C21;C22;C23withbasisfu)]TJ/F25 7.97 Tf 6.586 0 Td[(11;u)]TJ/F25 7.97 Tf 6.587 0 Td[(12;u)]TJ/F25 7.97 Tf 6.586 0 Td[(13;v)]TJ/F25 7.97 Tf 6.586 0 Td[(11;v)]TJ/F25 7.97 Tf 6.587 0 Td[(12;v)]TJ/F25 7.97 Tf 6.587 0 Td[(13g.Note 41

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heredimW1C21;C22;C23=5<6. Example3.10.5 TheCayleytableofZ2Z2representsthesecondofthetwomainclassofLatinsquaresoforder4:0BBBBB@12342143341243211CCCCCA:Withrespecttoanybasepointp,therearethreeinterleaved1-cycleCi1;Ci2;Ci3fori=1,2,3.ThestandardmoduleisthesumoftheprimarymodulePwhichhasbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(0p]];[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]];[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(2p]];[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(3p]];[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(4p]]g,W1C11;C12;C13withbasisfu11;u12;u13;v11;v12g,W1C21;C22;C23withbasisfu11;u12;u13;v11;v12g,andFvwithbasisv.The6elementsofthefourthsubconstituentarearrangedsothattwoappearineachof3rowsandineachof3columnsandtwohaveeachof3values.Formvasa1-linearcombinationofthecorrespondingcharacteristicvectorssothattheentrieswithacommonrow,column,orentryhavehaveoppositesign.NoteheredimW1C11;C12;C13=dimW1C21;C22;C23=5<6. 42

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4StronglyRegularGraphsfromaLatinSquareOnemaydeneseveralstronglyregulargraphsusingaLatinsquare.Inthischapterwedescribethelocalspectrumandsubconstituentalgebraofthesestronglyregulargraphs.Weusetheresultsofthepreviouschaptertodosoafterrecallingsomebackgroundmaterial. 4.1StronglyRegularGraphsWerecallsomefactsaboutstronglyregulargraphs.See[17,18]formoreinformation. Denition4.1.1 Anite,undirectedgraph)-296(withoutloopsormultipleedgesissaidtobestronglyregularSRGwithparameters;k;;ifithasmanyvertices,eachvertexhasexactlykmanyneighbors,anytwoadjacentverticeshaveexactlymanycommonneighbors,andanytwonon-adjacentverticeshaveexactlymanycommonneighbors.Astronglyregulargraphissaidtobetrivialwhen=0,inwhichcaseitisadisjointunionofcliquesofthesamesize.Let)-278(=X;Rbeanontrivialstronglyregulargraphwithparameters;k;;,andletA2MXdenotetheadjacencymatrixof)1(.ThenA0=I,A1=A,andA2=J)]TJ/F27 11.955 Tf 10.541 0 Td[(A)]TJ/F27 11.955 Tf 10.541 0 Td[(IaretheHadamardidempotentsofaBose-MesneralgebraM.TheprimitiveidempotentsofMarethemaximalprojectionsontotheeigenspacesofA.kisaneigenvalueofA;theothertwoeigenvaluesrandsaretherootsofthequadraticequation2+)]TJ/F27 11.955 Tf -442.009 -20.922 Td[(+)]TJ/F27 11.955 Tf 12.552 0 Td[(k=0andk>r>0,s)]TJ/F19 11.955 Tf 23.41 0 Td[(1.LetE0,E1,andE2denotetheprimitiveidempotentsassociatedwithk,r,ands,respectively.ThenE0,E1,andE2haverespectivemultiplicities1,mr=s+1kk)]TJ/F27 11.955 Tf 12.474 0 Td[(s=s)]TJ/F27 11.955 Tf 12.474 0 Td[(r,andms=)]TJ/F19 11.955 Tf 12.473 0 Td[(1)]TJ/F27 11.955 Tf 12.473 0 Td[(mr.ThedisjointunionofpmanycopiesofKqisatrivialSRGwithparameterspq;q)]TJ/F19 11.955 Tf 12.07 0 Td[(1;q)]TJ/F19 11.955 Tf 12.07 0 Td[(2;0.The 43

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eigenvaluesofthisgrapharekand)]TJ/F19 11.955 Tf 9.298 0 Td[(1withrespectivemultiplicitiespandpq)]TJ/F27 11.955 Tf 11.955 0 Td[(p.Recallthatthecomplementof)-342(isastronglyregulargraphwithparameters;)]TJ/F27 11.955 Tf -442.009 -20.922 Td[(k)]TJ/F19 11.955 Tf 10.666 0 Td[(1;)]TJ/F19 11.955 Tf 10.666 0 Td[(2k+)]TJ/F19 11.955 Tf 10.666 0 Td[(2;)]TJ/F19 11.955 Tf 10.666 0 Td[(2k+andeigenvalues)]TJ/F27 11.955 Tf 10.666 0 Td[(k)]TJ/F19 11.955 Tf 10.666 0 Td[(1,)]TJ/F27 11.955 Tf 9.299 0 Td[(s)]TJ/F19 11.955 Tf 10.666 0 Td[(1,and)]TJ/F27 11.955 Tf 9.298 0 Td[(r)]TJ/F19 11.955 Tf 10.666 0 Td[(1.However,)]TJ -380.005 -20.922 Td[(anditscomplementhavethesameBose-Mesneralgebraandsubconstituentalgebrawithrespecttoeachpoint.Werecallthesubconstituentalgebraofastronglyregulargraph[85]cf.[18]. Lemma4.1.2 [85]LetMdenotetheBose-Mesneralgebraofanontrivialstronglyreg-ulargraph)]TJ/F44 11.955 Tf 11.447 0 Td[(withparameters;k;;andeigenvaluesk,r,s.Fixabasepointp,andletTdenotethesubconstituentalgebraofMwithrespecttop.LetA=E1. i ThereisauniqueirreducibleT-moduleofdimensionthree.Ithasanorderedbasis0suchthat[A]0=0BBB@0k01k)]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 11.955 0 Td[(10k)]TJ/F27 11.955 Tf 11.955 0 Td[(1CCCA;[A]0=0BBB@k000r000s1CCCA:ThisistheprimarymodulePofT. ii Forpossiblymanydistinctnumbers62fr;sg,theremaybeanirreducibleT-modulewhichhasanorderedbasis1suchthat[A]1=0@r)]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F27 11.955 Tf 11.955 0 Td[(sr+s)]TJ/F27 11.955 Tf 11.955 0 Td[(1A;[A]1=0@r00s1A:WesaythatsuchamoduleisoftypeU. iii Theremaybeaone-dimensionalirreducibleT-moduleofsuchthat[A]1r=r;[A]1r=r:WesaythatsuchamoduleisoftypeU1;1r. 44

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iv Theremaybeaone-dimensionalirreducibleT-modulesuchthat[A]2s=s;[A]2s=s:WesaythatsuchamoduleisoftypeU2;2s. v Theremaybeaone-dimensionalirreducibleT-modulesuchthat[A]1s;r=s;[A]1s;r=r:WesaythatsuchamoduleisoftypeU2;1s;r. vi Theremaybeaone-dimensionalirreducibleT-modulesuchthat[A]2r;s=r;[A]2r;s=s:WesaythatsuchamoduleisoftypeU1;2r;s. vii Thasnootherpossibleirreduciblemodules. 4.2StronglyRegularGraphsfromaLatinSquare Lemma4.2.1 [5]WithNotation3.2.1, i EachofA1,A2,A3istheadjacencymatrixofstronglyregulargraphswithparame-tersn2;n)]TJ/F19 11.955 Tf 12.105 0 Td[(1;n)]TJ/F19 11.955 Tf 12.104 0 Td[(2;0andeigenvaluesn)]TJ/F19 11.955 Tf 12.105 0 Td[(1,)]TJ/F19 11.955 Tf 9.299 0 Td[(1withrespectivemultiplicitiesn,nn)]TJ/F19 11.955 Tf 10.072 0 Td[(1.ThesestronglyregulargraphsareisomorphicandsaidtobeoftypeLn;1.ThecomplementsofthesestronglyregulargraphshaverespectiveadjacencymatricesA4+A2+A3,A4+A1+A3,A4+A1+A2andparametersn2;n2)]TJ/F27 11.955 Tf 9.577 0 Td[(n;n2)]TJ/F19 11.955 Tf 9.577 0 Td[(2n+2;n2)]TJ/F27 11.955 Tf 9.577 0 Td[(n. ii EachofA1+A2,A1+A3,A2+A3istheadjacencymatrixofstronglyregulargraphswithparametersn2;2n)]TJ/F19 11.955 Tf 12.594 0 Td[(1;n)]TJ/F19 11.955 Tf 12.595 0 Td[(1;2andeigenvalues2n)]TJ/F19 11.955 Tf 12.595 0 Td[(1,n)]TJ/F19 11.955 Tf 12.595 0 Td[(2,)]TJ/F19 11.955 Tf 9.298 0 Td[(2withrespectivemultiplicities1,2n)]TJ/F19 11.955 Tf 12.776 0 Td[(1,andn)]TJ/F19 11.955 Tf 12.776 0 Td[(12.ThesestronglyregulargraphsareisomorphicandsaidtobeoftypeLn;2.ThecomplementsofthesestronglyregulargraphshaverespectiveadjacencymatricesA4+A3,A4+A2,A4+A1andparametersn2;n2)]TJ/F19 11.955 Tf 11.955 0 Td[(2n+1;n2)]TJ/F19 11.955 Tf 11.955 0 Td[(4n+6;n2)]TJ/F19 11.955 Tf 11.955 0 Td[(3n+3. 45

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iii A1+A2+A3istheadjacencymatrixofstronglyregulargraphswithparametersn2;3n)]TJ/F19 11.955 Tf 11.094 0 Td[(3;n;6andeigenvalues3n)]TJ/F19 11.955 Tf 11.094 0 Td[(3,n)]TJ/F19 11.955 Tf 11.094 0 Td[(3,and)]TJ/F19 11.955 Tf 9.299 0 Td[(3withrespectivemultiplicities1,3n)]TJ/F19 11.955 Tf 9.975 0 Td[(1,andn)]TJ/F19 11.955 Tf 9.975 0 Td[(2n)]TJ/F19 11.955 Tf 9.975 0 Td[(1.ThesestronglyregulargraphsareisomorphicandsaidtobeoftypeLn;3.ThecomplementofthisstronglyregulargraphshasadjacencymatrixA4andparametersn2;n2)]TJ/F19 11.955 Tf 11.955 0 Td[(3n+2;n2)]TJ/F19 11.955 Tf 11.955 0 Td[(6n+10;n2)]TJ/F19 11.955 Tf 11.955 0 Td[(5n+6. Denition4.2.2 WithNotation3.2.1,deneG1,G2,andG3tobethegraphswithvertexsetXandrespectiveadjacencymatricesA1,A1+A2,andA1+A2+A3.TheadjacencymatricesofGii=1;2;3areformedbyfusingsomeoftheHadamardidempotentsoftheBose-MesneralgebraofLcf.[9,76].Latinsquaresariseinthiscontextinaspecialway[56,57].IngeneraltherelationshipsbetweenthesubconstituentalgebrasofaBose-Mesneralgebraanditsfusionsisrathersubtle.ThusweshallconsideralltheSRG'sofLemma4.2.1asfusionsofM. 4.3FusionsTheadjacencyrelationofastronglyregulargraphassociatedwithaLatinsquareisformedbyfusingsomeoftherelationsR1,R2,andR3oftheassociationschemedenedbyaLatinsquare.Wecommentonthisperspective. Notation4.3.1 LetXdenoteanitenonemptyset.LetNandMdenotee+1-andd+1-dimensionalBose-MesneralgebrasonX,respectively.LetfBigei=0andfAigdi=0denotetherespectiveHadamardidempotentsofNandM,andletfFigei=0andfEigdi=0denotetherespectiveprimitiveidempotentsofNandM.Fixp2X.LetfBigei=0andfAigdi=0denotetherespectivedualHadamardidempotentsofNandMwithrespecttop,andletfFigei=0andfEigdi=0denotetherespectivedualidempotentsofNandMwithrespecttop.LetSandTdenotetherespectivesubconstituentalgebrasofNandMwithrespecttop. Theorem4.3.2 [9,76]WithNotation4.3.1,letPandQdenotetheeigenmatrixanddualeigenmatrixofM,respectively.Thenthefollowingareequivalent. i NisaBose-MesnersubalgebraofM. 46

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ii Thereexistsapairofpartitions=f0=f0g;1;:::;dgand0=f00=f0g;01;:::;0dgoff0;1;:::;egsuchthatforallh,ih;idBh=Xh02hAh0;tBh=X`02`A`0forsome``d;Xh02hPh0;j=Xh02hPh0;kforallj;k20i. iii Thereexistsapairofpartitions=f0=f0g;1;:::;dgand0=f00=f0g;01;:::;0dgoff0;1;:::;egsuchthatforallh,ih;idFh=Xh020hEh0;tFh=X`020`E`0forsome``d;Xh020hQh0;j=Xh020hQh0;kforallj;k2i.Supposei{iiihold.Thenthepartitionsiniiandiiicoincide,andthepartitions0iniiandiiicoincide.Moreoveruniquelydetermines0andviceversa.WerefertotheBose-MesnersubalgebraMasthe;0-fusionofN. Corollary4.3.3 WithNotation4.3.1,assumethatNisthe;0-fusionofM.Thenthefollowinghold. i Fi=Ph02hEiid. ii Bi=Ph020hAiid.Proof.ClearfromTheorem4.3.2sinceislinear.2 Lemma4.3.4 WithNotation4.3.1,assumethatNisthe;0-fusionofM.TheneveryT-moduleisanS-modules.Proof.ClearfromTheorem4.3.2andCorollary4.3.3.2DespiteLemma4.3.4,theisomorphismclassesofirreducibleS-modulesarenotob-viouslydeterminedbythoseofirreducibleT-modules.Forexample,anirreducibleT-modulemaydecomposefurtherintothesumofirreducibleS-modulesthisisnecessarily 47

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thecasefortheprimarymodule.Anothercomplicationisthepossibilitythatnon-isomorphicT-modulesareisomorphicasS-modules.WeshallseethatthisisthecaseforWandW.Latinsquaresarecloselyrelatedtoaratherinterestingpropertyconcerningfusions. Denition4.3.5 Ane-dimensionalBose-Mesneralgebraissaidtobeamorphouswhen-everforanypartition=f0=f0g;1;:::;dgoff0;1;:::;eg,thereisapartition0=f00=f0g;01;:::;0dgoff0;1;:::;egsuchthatthereisaBose-MesnersubalgebraofMwhichisthe;0-fusionofM. Denition4.3.6 ABose-MesneralgebraissaidtobeofnegativeLatinsquaretypewheneverevery0;1-matrixwithzerodiagonalistheadjacencymatrixofstronglyregulargraphwith=ss+1respectively=rr+1,wherethestronglyregulargraphhaseigenvaluesk>s>r. Theorem4.3.7 [56,57]ABose-MesneralgebraisamorphousifandonlyifitisofLatinsquaretypeornegativeLatinsquaretype. 4.4G3TousethetheoryofSection3.4,wewillneedtodistinguishtheobjectswhicharisefromtheBose-MesneralgebraofaLatinsquareandthosewhicharisefromtheseSRG's. Notation4.4.1 LetLdenotetheLatinsquareofordern5.LetXbethesetfi;j;Li;jj1i;jng.Dene)]TJ/F28 7.97 Tf 53.096 -1.793 Td[(ii4relativetotherelationsinequations.1.1-.1.5.LetMdenotetheBose-MesneralgebraofL,andletfAig4i=0denotetheHadamardidempotentsofM.Fixabasepointp2X.LetfEig4i=0denotethedualidempotentsandletTdenotethesubconstituentalgebraofMwithrespecttop.WenowdescribetheirreduciblemodulesandlocalspectrumofG3. Notation4.4.2 WithreferencetoNotation4.4.1,letG3beasinDenition4.2.2.LetfBig2i=0betheHadamardidempotentsoftheBose-MesneralgebraNofG3,whereB=B1istheadjacencymatrixofG3.LetFi=BibethedualidempotentsofNwith 48

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respecttop,andletSdenotethesuconstituentalgebraofNwithrespecttop.LetN3betheneighborsofpinG3. Lemma4.4.3 WithNotation4.4.1and4.4.2,thefollowinghold. i N3=)]TJ/F25 7.97 Tf 19.74 -1.793 Td[(1p[)]TJ/F25 7.97 Tf 7.315 -1.793 Td[(2p[)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(3p. ii )]TJ/F25 7.97 Tf 7.314 -1.793 Td[(1p,)]TJ/F25 7.97 Tf 7.315 -1.793 Td[(2p,and)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(3paremutuallydisjointandtheinducedsubgraphsofGon)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(1p,)]TJ/F25 7.97 Tf 7.315 -1.793 Td[(2p,and)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(3paren)]TJ/F19 11.955 Tf 11.956 0 Td[(1-cliques. iii LetbethesubgraphoftheinducedsubgraphofG3onN3formedbyremovingtheedgeswithbothendpointsinR1,bothendpointinR2,orbothendpointinR3.Thenisadisjointunionofcycles:Foreachinterleavedtripleofk-cyclesC1,C2,C3asinLemma3.2.5thefollowingpointsforma3k-cyclein:rp;c1;e1,r1;cp;e1,r1;c2;ep,rp;c2;e2,r2;cp;e2,r2;c3;ep,...,rp;ck;ek,rk;cp;ek,rk;c1;ep,rp;c1;e1.Proof.Theneighborsofp=pr;pc;peshareacommonentrywithp,soN3=)]TJ/F25 7.97 Tf 20.172 -1.793 Td[(1p[)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(2p[)]TJ/F25 7.97 Tf 7.315 -1.793 Td[(3p.Each)]TJ/F28 7.97 Tf 49.313 -1.793 Td[(ipisacliqueofsizen)]TJ/F19 11.955 Tf 11.788 0 Td[(1byconstruction.Thisunionisdisjointsinceanytwoentriesuniquelydeterminethethird.PartiiifollowsfromLemma3.2.5seealso[35].2 Lemma4.4.4 WithNotation4.4.1and4.4.2,theprimaryT-moduleisthedirectsumof3irreducibleS-modules: i TheprimaryS-modulewithbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(0p]],[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]]+[[)]TJ/F25 7.97 Tf 35.306 -1.793 Td[(2p]]+[[)]TJ/F25 7.97 Tf 35.306 -1.793 Td[(3p]],[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(4p]]g. ii AmoduleoftypeU1;1n)]TJ/F19 11.955 Tf 11.955 0 Td[(3withbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]])]TJ/F19 11.955 Tf 11.955 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(3p]]g. iii AmoduleoftypeU1;1n)]TJ/F19 11.955 Tf 11.955 0 Td[(3withbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(2p]])]TJ/F19 11.955 Tf 11.955 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(3p]]g.Proof.i:Clear.ii:Let~BbetheprincipalminorofF1BF1inducedbyelementsofN3.Observethat~Bistheadjacencymatrixoftheinducedsubgraph)-363(ontheneighborsofp.Firstweclaimthat[[)]TJ/F25 7.97 Tf 88.169 -1.793 Td[(1p]])]TJ/F19 11.955 Tf 12.53 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(3p]]and[[)]TJ/F25 7.97 Tf 49.239 -1.793 Td[(2p]])]TJ/F19 11.955 Tf 12.53 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(3p]]arelinearlyindependentvectors 49

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intheeigenspaceof~Bassociatedwithn)]TJ/F19 11.955 Tf 12.163 0 Td[(3.Eachvertexof)]TJ/F25 7.97 Tf 101.323 -1.793 Td[(1pisadjacentton)]TJ/F19 11.955 Tf 12.162 0 Td[(2verticesof)]TJ/F25 7.97 Tf 65.424 -1.793 Td[(1p,eachvertexof)]TJ/F25 7.97 Tf 99.321 -1.793 Td[(2pisadjacenttoexactlyonevertexof)]TJ/F25 7.97 Tf 208.992 -1.793 Td[(1p,andeachvertexof)]TJ/F25 7.97 Tf 84.296 -1.793 Td[(3pisadjacenttoexactlyonevertexof)]TJ/F25 7.97 Tf 204.21 -1.793 Td[(1p.Thus~B[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]]=n)]TJ/F19 11.955 Tf -442.009 -20.922 Td[(2[[)]TJ/F25 7.97 Tf 22.231 -1.793 Td[(1p]]+[[)]TJ/F25 7.97 Tf 32.745 -1.793 Td[(2p]]+[[)]TJ/F25 7.97 Tf 32.745 -1.793 Td[(3p]].Similarly,~B[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(2p]]=n)]TJ/F19 11.955 Tf 10.674 0 Td[(2[[)]TJ/F25 7.97 Tf 22.23 -1.793 Td[(2p]]+[[)]TJ/F25 7.97 Tf 32.744 -1.793 Td[(1p]]+[[)]TJ/F25 7.97 Tf 32.744 -1.793 Td[(3p]]and~B[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(3p]]=n)]TJ/F19 11.955 Tf 10.089 0 Td[(2[[)]TJ/F25 7.97 Tf 22.23 -1.794 Td[(3p]]+[[)]TJ/F25 7.97 Tf 31.574 -1.794 Td[(1p]]+[[)]TJ/F25 7.97 Tf 31.574 -1.794 Td[(2p]].Thus[[)]TJ/F25 7.97 Tf 57.779 -1.794 Td[(1p]])]TJ/F19 11.955 Tf 10.088 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(3p]]and[[)]TJ/F25 7.97 Tf 45.725 -1.794 Td[(2p]])]TJ/F19 11.955 Tf 10.089 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(3p]]areeigenvectorsfor~Bassociatedwithn)]TJ/F19 11.955 Tf 11.503 0 Td[(3.Theyareclearlylinearlyindependentsincetheyhavesomenonzeroentriesinmutuallydistinctpositions.2 Theorem4.4.5 WithNotation4.4.1and4.4.2,assumen5.LetC1,C2,C3beaninterleavedtripleofk-cycles. i W1C1;C2;C3isthedirectsumof3irreducibleS-modules: a AmoduleoftypeUwithbasisfu11+u12+u13,v11+v12+v13g. b AmoduleoftypeU)]TJ/F19 11.955 Tf 9.298 0 Td[(2withbasisfu11)]TJ/F19 11.955 Tf 11.955 0 Td[(2u12+u13,v11)]TJ/F19 11.955 Tf 11.955 0 Td[(2v12+v13g. c AmoduleoftypeU)]TJ/F19 11.955 Tf 9.298 0 Td[(2withbasisfu11+2u12)]TJ/F19 11.955 Tf 11.955 0 Td[(3u13,v11+2v12)]TJ/F19 11.955 Tf 11.955 0 Td[(3v13g. ii Ifkiseven,thenW)]TJ/F25 7.97 Tf 6.587 0 Td[(1C1;C2;C3isthedirectsumof4irreducibleS-modules: a AmoduleoftypeUwithbasisfu)]TJ/F25 7.97 Tf 6.587 0 Td[(11+2u)]TJ/F25 7.97 Tf 6.586 0 Td[(12+u)]TJ/F25 7.97 Tf 6.586 0 Td[(13,v)]TJ/F25 7.97 Tf 6.586 0 Td[(11+v)]TJ/F25 7.97 Tf 6.586 0 Td[(13g. b AmoduleoftypeUwithbasisfu)]TJ/F25 7.97 Tf 6.587 0 Td[(11)]TJ/F27 11.955 Tf 11.955 0 Td[(u)]TJ/F25 7.97 Tf 6.587 0 Td[(12)]TJ/F19 11.955 Tf 11.955 0 Td[(2u)]TJ/F25 7.97 Tf 6.587 0 Td[(13,v)]TJ/F25 7.97 Tf 6.587 0 Td[(11)]TJ/F27 11.955 Tf 11.955 0 Td[(v)]TJ/F25 7.97 Tf 6.587 0 Td[(12g. c AmoduleoftypeU1;2)]TJ/F19 11.955 Tf 9.298 0 Td[(3;n)]TJ/F19 11.955 Tf 11.955 0 Td[(3withbasisfu)]TJ/F25 7.97 Tf 6.587 0 Td[(11)]TJ/F27 11.955 Tf 11.955 0 Td[(u)]TJ/F25 7.97 Tf 6.586 0 Td[(12+u)]TJ/F25 7.97 Tf 6.586 0 Td[(13g. d AmoduleoftypeU2;1n)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.299 0 Td[(3withbasisfv)]TJ/F25 7.97 Tf 6.587 0 Td[(11+v)]TJ/F25 7.97 Tf 6.587 0 Td[(12)]TJ/F27 11.955 Tf 11.955 0 Td[(v)]TJ/F25 7.97 Tf 6.587 0 Td[(13g. iii Foreachrootofunity62f1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1gwithorderwhichdividesk,letii=1;2;3betherootsofthepolynomialx3+3x2)]TJ/F19 11.955 Tf 12.717 0 Td[(21+Re.ThenWC1;C2;C3istheorthogonaldirectsumof3irreducibleS-modules:Fori=1,2,3thereisamoduleoftypeUiwithbasisfu1+biu2+ciu3,bi+civ1++)]TJ/F25 7.97 Tf 6.587 0 Td[(1civ2++biv3g,wherebi=1++i i2+2iandci=1++i i2+2i.Proof.iii:Letu=u1+bu2+cu3beaneigenvectorofF1B1F1witheigenvalue,ie,F1B1F1u=u1+bu2+cu3.ExpandingF1=E1+E2+E3andB1=A1+A2+A3 50

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anddecomposingintosubconstituentsgivesE1A1E1+E2A3E1+E3A2E1u1)]TJ/F27 11.955 Tf 11.956 0 Td[(u1=0;E2A2E2+E1A3E2+E3A1E2bu2)]TJ/F27 11.955 Tf 11.956 0 Td[(bu2=0;E3A3E3+E1A2E3+E2A1E3cu3)]TJ/F27 11.955 Tf 11.955 0 Td[(cu3=0:ByFigures3.6{3.9,)]TJ/F19 11.955 Tf 9.298 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(+b+)]TJ/F25 7.97 Tf 6.587 0 Td[(1c=1+)]TJ/F19 11.955 Tf 9.298 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(b+c=+b+)]TJ/F19 11.955 Tf 9.299 0 Td[(1)]TJ/F27 11.955 Tf 11.956 0 Td[(c=0:.4.1Byelementarylinearalgebratheseequationsarelinearlyindependentunless=1,whichdoesn'toccuriniii.Eliminatingbandcgivesthatisarootofthecubicequationx3+3x2)]TJ/F19 11.955 Tf 12.801 0 Td[(2Re+1=0.Notethat)]TJ/F19 11.955 Tf 9.298 0 Td[(1Re1,sothisequationhas3realrootsbetween1and-3.Theyaredistinctsince6=1.Writei,i=1;2;3todenotetheseroots.Eachisassociatedwithaneigenvectorsay,uiforF1B1F1asdescribedinthestatement.ApplyingF2B1F1touiforsomei=1;2;3gives,vi=bi+civ1++civ2++biv3.Foreachi=1;2;3,thesetofvectorsfui;vigislinearlyindependent.ItisclosedundertheactionofallothergeneratorsofS,anditisasubsetoftheT-module,WC1;C2;C3.Souiandvispanatwo-dimensionalS-submodule.ObservethatthismoduleisoftypeUi.Notethatfori6=j,UiandUjarenotisomorphicsincethei6=j.i:For=1,thesystemofequations.4.1becomes:)]TJ/F19 11.955 Tf 9.299 0 Td[(1)]TJ/F27 11.955 Tf 11.956 0 Td[(+b+c=1)]TJ/F19 11.955 Tf 11.955 0 Td[(+b+c=1+b)]TJ/F19 11.955 Tf 11.956 0 Td[(+c=0:Thissystemofequationshasnosolutionif=0.Alsoif=)]TJ/F19 11.955 Tf 9.299 0 Td[(2,thenb=)]TJ/F19 11.955 Tf 9.298 0 Td[(1)]TJ/F27 11.955 Tf 12.205 0 Td[(candtherearetwolinearlyindependenteigenvectorsassociatedwiththeeigenvalue-2asdescribedinib,c.If=2f)]TJ/F19 11.955 Tf 28.333 0 Td[(2;0g,thenhastosatisfythequadraticequation:x2+x)]TJ/F19 11.955 Tf 11.987 0 Td[(2,ie=1.Inthiscase,c=1andb=1.Henceu11+u12+u13isaneigenvectorofF1B1F1withaneigenvalue=1.NowF2B1F1u11+u12+u13=v11+v12+v13,andallotheractionsonthisvectorarelinearcombinationsofthesetwovectors.Moreover,thesetwovectorsarelinearlyindependent,sotheyformabasisforanS-submoduleof 51

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W1C1;C2;C3.Similarlyu11)]TJ/F19 11.955 Tf 11.057 0 Td[(2u12+u13andv11)]TJ/F19 11.955 Tf 11.057 0 Td[(2v12+v13formabasisforanS-submoduleofW1C1;C2;C3.Alsou11+2u12)]TJ/F19 11.955 Tf 10.961 0 Td[(3u13andv11+2v12)]TJ/F19 11.955 Tf 10.961 0 Td[(3v13formabasisforanS-submoduleofW1C1;C2;C3.ii:For=)]TJ/F19 11.955 Tf 9.298 0 Td[(1,thesystemofequations4.4.1becomes:)]TJ/F19 11.955 Tf 9.298 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(+b)]TJ/F27 11.955 Tf 11.955 0 Td[(c=1)]TJ/F19 11.955 Tf 11.956 0 Td[(+b+c=)]TJ/F19 11.955 Tf 9.299 0 Td[(1+b)]TJ/F19 11.955 Tf 11.955 0 Td[(+c=0:If6=0,thenb=)]TJ/F19 11.955 Tf 9.298 0 Td[(1,c=1and=)]TJ/F19 11.955 Tf 9.298 0 Td[(3.SinceisaneigenvalueofB,u)]TJ/F25 7.97 Tf 6.587 0 Td[(11)]TJ/F27 11.955 Tf 11.627 0 Td[(u)]TJ/F25 7.97 Tf 6.587 0 Td[(12+u)]TJ/F25 7.97 Tf 6.587 0 Td[(13isabasisforaone-dimensionalS-submoduleofW)]TJ/F25 7.97 Tf 6.586 0 Td[(1C1;C2;C3.If=0,thenb=c+1whichgivestwolinearlyindependenteigenvectorsu)]TJ/F25 7.97 Tf 6.586 0 Td[(11+2u)]TJ/F25 7.97 Tf 6.586 0 Td[(12+u)]TJ/F25 7.97 Tf 6.587 0 Td[(13andu)]TJ/F25 7.97 Tf 6.586 0 Td[(11)]TJ/F27 11.955 Tf 11.033 0 Td[(u)]TJ/F25 7.97 Tf 6.587 0 Td[(12)]TJ/F19 11.955 Tf 11.032 0 Td[(2u)]TJ/F25 7.97 Tf 6.586 0 Td[(13associatedwith=0.Asini,u)]TJ/F25 7.97 Tf 6.586 0 Td[(11+2u)]TJ/F25 7.97 Tf 6.586 0 Td[(12+u)]TJ/F25 7.97 Tf 6.587 0 Td[(13andv)]TJ/F25 7.97 Tf 6.587 0 Td[(11+v)]TJ/F25 7.97 Tf 6.587 0 Td[(13formabasisforatwo-dimensionalS-submoduleofW)]TJ/F25 7.97 Tf 6.586 0 Td[(1C1;C2;C3,andu)]TJ/F25 7.97 Tf 6.586 0 Td[(11)]TJ/F27 11.955 Tf 12.481 0 Td[(u)]TJ/F25 7.97 Tf 6.586 0 Td[(12)]TJ/F19 11.955 Tf 12.481 0 Td[(2u)]TJ/F25 7.97 Tf 6.587 0 Td[(13andv)]TJ/F25 7.97 Tf 6.587 0 Td[(11)]TJ/F27 11.955 Tf 12.482 0 Td[(v)]TJ/F25 7.97 Tf 6.586 0 Td[(12formbasisforatwo-dimensionalirreducibleS-submoduleofW)]TJ/F25 7.97 Tf 6.586 0 Td[(1C1;C2;C3.NotethatF2B1F2v)]TJ/F25 7.97 Tf 6.586 0 Td[(11+v)]TJ/F25 7.97 Tf 6.586 0 Td[(12)]TJ/F27 11.955 Tf 12.311 0 Td[(v)]TJ/F25 7.97 Tf 6.587 0 Td[(13=4v)]TJ/F25 7.97 Tf 6.586 0 Td[(11+v)]TJ/F25 7.97 Tf 6.586 0 Td[(12)]TJ/F27 11.955 Tf 12.311 0 Td[(v)]TJ/F25 7.97 Tf 6.586 0 Td[(13,andallotheractionsonthisvectoris0.Hencev)]TJ/F25 7.97 Tf 6.586 0 Td[(11+v)]TJ/F25 7.97 Tf 6.586 0 Td[(12)]TJ/F27 11.955 Tf 11.346 0 Td[(v)]TJ/F25 7.97 Tf 6.586 0 Td[(13formsabasisforaone-dimensionalirreducibleS-submoduleofW)]TJ/F25 7.97 Tf 6.587 0 Td[(1C1;C2;C3.2 Lemma4.4.6 WithNotation4.4.1and4.4.2,foreachv2E4VwhichisorthogonaltoallirreducibleT-modulesconstructedviaLemma3.4.2andLemma3.4.4,Fvisaone-dimensionalirreducibleS-moduleoftypeU2;2)]TJ/F19 11.955 Tf 9.299 0 Td[(3. Corollary4.4.7 WithNotation4.4.1and4.4.2,thespectrumoftheinducedsubgraphofG3onN3isdeterminedasfollows. i Themodulesoflemma4.4.4contribute1tothemultiplicityof3n)]TJ/F19 11.955 Tf 11.079 0 Td[(1and2tothemultiplicityofn)]TJ/F19 11.955 Tf 11.955 0 Td[(3. ii LetC1;C2;C3beaninterleavedk-cycle.Then a C1;C2;C3contributesonetothemultiplicityof1andtwotothemultiplicityof-2. b Ifkiseven,thenC1;C2;C3contributestwotothemultiplicityof0andonetothemultiplicityof-3. 52

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c Foreachkthrootofunity6=1,C1;C2;C3contributesonetothemultiplicityofeachrootofx3+3x2)]TJ/F19 11.955 Tf 11.956 0 Td[(21+Re.Proof.ThesearetheeigenvaluesofF1BF1oneachirreducibleS-module.2sinceisontheIfn7,thennointerleavedcyclecontributestothemultiplicityofn)]TJ/F19 11.955 Tf 12.294 0 Td[(3.Ifn=6,thena4-cyclecontributestwotothemultiplicityofthen)]TJ/F19 11.955 Tf 12.294 0 Td[(3=3.Ifn=5,then=1contributesonetothemultiplicityofn)]TJ/F19 11.955 Tf 12.438 0 Td[(3=2.Ifn4,thennointerleavedcyclecontributestothemultiplicityofn)]TJ/F19 11.955 Tf 12.437 0 Td[(3.NotethatWC1;C2;C3andWC1;C2;C3areisomorphicasS-modulessincetheirreduciblesthattheydecomposeintoareisomorphic.Inparticular,theyhavethesamecontributiontothelocalspectrum. 4.5G2WenowreportresultsforG2withoutproofs. Notation4.5.1 WithreferencetoNotation4.4.1,letG2beasinDenition4.2.2.LetNbetheBose-MesneralgebraofG2,andletSdenotethesuconstituentalgebraofNwithrespecttop.LetN2betheneighborsofpinG2. Lemma4.5.2 WithNotation4.4.1and4.5.1,N2=)]TJ/F25 7.97 Tf 19.97 -1.793 Td[(1p[)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(2p,)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(1pand)]TJ/F25 7.97 Tf 7.315 -1.793 Td[(2paredisjoint,andtheinducedsubgraphsofG2on)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(1pand)]TJ/F25 7.97 Tf 7.314 -1.793 Td[(2paren)]TJ/F19 11.955 Tf 11.955 0 Td[(1-cliques. Lemma4.5.3 WithNotation4.4.1and4.5.1,theprimaryT-moduleistheorthogonaldirectsumof3irreducibleS-modules: i TheprimaryS-modulewithbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(0p]],[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(1p]]+[[)]TJ/F25 7.97 Tf 35.306 -1.794 Td[(2p]],[[)]TJ/F25 7.97 Tf 11.825 -1.794 Td[(3p]]+[[)]TJ/F25 7.97 Tf 35.306 -1.794 Td[(4p]]g. ii AmoduleoftypeU1;1n)]TJ/F19 11.955 Tf 11.955 0 Td[(2withbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]])]TJ/F19 11.955 Tf 11.955 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(2p]]g. iii AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(2withbasisfn)]TJ/F19 11.955 Tf 11.955 0 Td[(2[[)]TJ/F25 7.97 Tf 22.23 -1.793 Td[(3p]])]TJ/F19 11.955 Tf 11.955 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(4p]]g. Theorem4.5.4 Assumen5.WithNotation4.4.1and4.5.1,letC1,C2,C3beaninterleavedtripleofk-cycles. i W1C1;C2;C3isthedirectsumof4irreducibleS-modules: 53

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a AmoduleoftypeU)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisfu11+u12,v11+v12+2u13g. b AmoduleoftypeU)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisfu11)]TJ/F27 11.955 Tf 11.955 0 Td[(u12,v12)]TJ/F27 11.955 Tf 11.955 0 Td[(v11g. c AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(2withbasisfv13+u13g. d AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(2withbasisfv11+v12+v13)]TJ/F19 11.955 Tf 11.955 0 Td[(n)]TJ/F19 11.955 Tf 11.956 0 Td[(4u13g. ii Ifkiseven,thenW)]TJ/F25 7.97 Tf 6.586 0 Td[(1C1;C2;C3isthedirectsumof4irreducibleS-modules: a AmoduleoftypeU)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisfu)]TJ/F25 7.97 Tf 6.586 0 Td[(11+u)]TJ/F25 7.97 Tf 6.587 0 Td[(12,v)]TJ/F25 7.97 Tf 6.587 0 Td[(11+v)]TJ/F25 7.97 Tf 6.587 0 Td[(12g. b AmoduleoftypeU)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisfu)]TJ/F25 7.97 Tf 6.586 0 Td[(11)]TJ/F27 11.955 Tf 11.955 0 Td[(u)]TJ/F25 7.97 Tf 6.586 0 Td[(12,v)]TJ/F25 7.97 Tf 6.586 0 Td[(12)]TJ/F27 11.955 Tf 11.956 0 Td[(v)]TJ/F25 7.97 Tf 6.586 0 Td[(11)]TJ/F19 11.955 Tf 11.955 0 Td[(2u)]TJ/F25 7.97 Tf 6.586 0 Td[(13g. c AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(2withbasisfv)]TJ/F25 7.97 Tf 6.586 0 Td[(12)]TJ/F27 11.955 Tf 11.955 0 Td[(v)]TJ/F25 7.97 Tf 6.587 0 Td[(11+n)]TJ/F19 11.955 Tf 11.955 0 Td[(3u)]TJ/F25 7.97 Tf 6.586 0 Td[(13g. d AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(2withbasisfv)]TJ/F25 7.97 Tf 6.586 0 Td[(11+v)]TJ/F25 7.97 Tf 6.586 0 Td[(12+n)]TJ/F19 11.955 Tf 11.955 0 Td[(1v)]TJ/F25 7.97 Tf 6.586 0 Td[(13g. iii Foreachrootofunity62f1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1gwithorderwhichdividesk,WC1;C2;C3isthedirectsumof4irreducibleS-modules: a AmoduleoftypeU)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisfu1+u2,v1+v2++u3g. b AmoduleoftypeU)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisfu1)]TJ/F27 11.955 Tf 11.955 0 Td[(u2,v2)]TJ/F27 11.955 Tf 11.955 0 Td[(v1+)]TJ/F19 11.955 Tf 11.955 0 Td[(1u3g. c AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(2withbasisf+1u3+v1+v2+n)]TJ/F19 11.955 Tf 11.955 0 Td[(1v3g. d AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(2withbasisf+2+2n)]TJ/F19 11.955 Tf 11.672 0 Td[(2u3+)]TJ/F27 11.955 Tf 11.671 0 Td[(v1+)]TJ/F19 11.955 Tf -390.736 -20.921 Td[(1v2++n)]TJ/F19 11.955 Tf 11.955 0 Td[(1v3g. Lemma4.5.5 WithNotation4.4.1and4.5.1,foreachv2E4VwhichisorthogonaltoallirreducibleT-modulesconstructedviaLemma3.4.2andLemma3.4.4,Fvisaone-dimensionalirreducibleS-moduleoftypeU2;2)]TJ/F19 11.955 Tf 9.299 0 Td[(2. Corollary4.5.6 WithNotation4.4.1and4.5.1,thespectrumoftheinducedsubgraphofG2onN2isn)]TJ/F19 11.955 Tf 11.955 0 Td[(22,)]TJ/F19 11.955 Tf 9.299 0 Td[(12n)]TJ/F25 7.97 Tf 6.586 0 Td[(2. 4.6G1WereportresultsforG1withoutproofs. Notation4.6.1 WithreferencetoNotation4.4.1,letG1beasinDenition4.2.2.LetNbetheBose-MesneralgebraofG1,andletSdenotethesuconstituentalgebraofNwithrespecttop.LetN1betheneighborsofpinG1. 54

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Lemma4.6.2 WithNotation4.4.1and4.6.1,N1=)]TJ/F25 7.97 Tf 19.929 -1.793 Td[(1p,andtheinducedsubgraphofG1onN1isann)]TJ/F19 11.955 Tf 11.955 0 Td[(1-clique. Lemma4.6.3 WithNotation4.4.1and4.6.1,theprimaryT-moduleistheorthogonaldirectsumof3irreducibleS-modules: i TheprimaryS-modulewithbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(0p]],[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(1p]],[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(2p]]+[[)]TJ/F25 7.97 Tf 35.306 -1.793 Td[(3p]]+[[)]TJ/F25 7.97 Tf 35.306 -1.793 Td[(4p]]g. ii AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisf[[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(2p]])]TJ/F19 11.955 Tf 11.955 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(3p]]g. iii AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisfn)]TJ/F19 11.955 Tf 11.955 0 Td[(2[[)]TJ/F25 7.97 Tf 22.23 -1.793 Td[(3p]])]TJ/F19 11.955 Tf 11.955 0 Td[([[)]TJ/F25 7.97 Tf 11.825 -1.793 Td[(4p]]g. Theorem4.6.4 Assumen5.WithNotation4.4.1and4.6.1,letC1,C2,C3beaninter-leavedtripleofk-cycles.Foreachrootofunitywithorderwhichdividesk,WC1;C2;C3istheorthogonaldirectsumof6irreducibleS-modules: i AmoduleoftypeU1;1)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisfu1g. ii AmoduleoftypeU2;2n)]TJ/F19 11.955 Tf 11.955 0 Td[(1withbasisfu2+u3+v1g. iii AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisfu2+v3g. iv AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisfu2+v2g. v AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisf)]TJ/F27 11.955 Tf 11.956 0 Td[(nu2+v1g. vi AmoduleoftypeU2;2)]TJ/F19 11.955 Tf 9.298 0 Td[(1withbasisf)]TJ/F27 11.955 Tf 15.276 0 Td[(u2+u3g. Lemma4.6.5 WithNotation4.4.1and4.6.1,foreachv2E4VwhichisorthogonaltoallirreducibleT-modulesconstructedviaTheorems2.3.8,??,and??,Fvisaone-dimensionalirreducibleS-moduleoftypeU2;2)]TJ/F19 11.955 Tf 9.299 0 Td[(1. Corollary4.6.6 WithNotation4.4.1and4.6.1,thespectrumoftheinducedsubgraphofG1onN1isn)]TJ/F19 11.955 Tf 11.955 0 Td[(2,)]TJ/F19 11.955 Tf 9.298 0 Td[(1n)]TJ/F25 7.97 Tf 6.586 0 Td[(2. 55

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5IsomorphismsInthischapterwecompareseveralnotionsofisomorphismforsubconstituentalgebras.Latinsquaresprovidearichsourceofexamplesandcounter-examples. 5.1IsomorphismsofBose-MesnerAlgebraWerecallsomenotionsofisomorphismforBose-Mesneralgebras.See[5,41]formoredetails.Forcompletenessweprovideproofsofsomebasicresults. Denition5.1.1 LetMandM0denoteBose-MesneralgebrasonXandX0,respec-tively. i MandM0aresaidtobealgebraicallyisomorphicifthereexistsalinearbijection:M!M0suchthatAB=ABandAB=ABforallA,B2M. ii MandM0aresaidtobecombinatoriallyisomorphicifthereexistsabijection:X!X0suchthatM0=fA02MX0jforsomeA2M;A0x;y=Ax;y8x;y2Xg. Lemma5.1.2 WithreferencetoDenition5.1.1,ifMandM0arecombinatoriallyisomorphic,thentheyarealgebraicallyisomorphic.Proof.WemayrepresentthecombinatorialisomophisminducedbybyconjugationbyapermutationmatrixP2MX;X0.ThenclearlyforallA,B2M,P)]TJ/F25 7.97 Tf 6.587 0 Td[(1ABP=P)]TJ/F25 7.97 Tf 6.587 0 Td[(1APP)]TJ/F25 7.97 Tf 6.586 0 Td[(1BPandP)]TJ/F25 7.97 Tf 6.586 0 Td[(1ABP=P)]TJ/F25 7.97 Tf 6.586 0 Td[(1APP)]TJ/F25 7.97 Tf 6.587 0 Td[(1BP.ThusconjugationbyPdenesanalgebraicisomorphism.2 56

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Lemma5.1.3 WithreferencetoDenition5.1.1,ifMandM0arecombinatoriallyisomorphic,thenthereexistsabijection:f0;1;:::;dg!f0;1;:::;d0gsuchthatA0ix;y=Aix;y,wherefAigdi=0andfA0igd0i=0denotetherespectiveHadamardidempotentsofMandM0.Proof.ObservethatthesetfB0igdi=0denedbyB0ix;y=Aix;yforallx,y2Xsatises.1.1{.1.3,soitisthebasisofHadamardidempotentsofM0.TheresultfollowsfromtheuniquenessofthebasisofHadamardidempotentsofaBose-Mesneralgebra.2 Theorem5.1.4 [10]TwoBose-MesneralgebrasarealgebraicallyisomorphicifandonlyiftheyhavethesameintersectionnumbersrelativetosomeorderingsoftheHadamardidempotents.Proof.Denealinearmapf:M!M0takingAi2MtoA0i2M0.NowfAiAj=fPkpkijAk=PkpkijA0k,andfAifAj=A0iA0j=Pkpk0ijA0k,andsofAiAj=fAifAjifandonlyifpkij=pk0ij.NotethatfAiAj=ijfAi=ijA0i=A0iA0i=fA0ifA0i.2TodiscusscombinatorialisomorphismofBose-Mesneralgebraswerecallcommutativeassociationschemeandtheirisomorphisms. Denition5.1.5 Ad-classcommutativeassociationschemeisapairX;fRigdi=0,whereXisanitenon-emptysetandtheRiarerelationsonXXsuchthat i R0=fx;xjx2Xg; ii forallx,y2X,thereexistsaniidsuchthatx;y2Ri; iii foralliidthereexistsi0i0dsuchthatx;y2Riifandonlyify;x2Ri0; iv forallh,i,jh;i;jdthereexistsascalarphijsuchthatforallx;y2Rh,jfz2Xjx;z2Ri;z;y2Rjgj=phij. v phij=phjiforalli,j,ki;j;kd. 57

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Denition5.1.6 TwoassociationschemesX;fRigdi=0andX0;fR0igd0i=0areisomor-phicwheneverthereexistbijections:X!X0and:f0;1;2;:::;dg!f0;1;2;:::;d0gsuchthatxRiyifandonlyifxR0iyforallx,y2X. Lemma5.1.7 [5,10,17,41] i LetX;fRigdi=0denotead-classcommutativeassociationscheme.Thenthema-tricesAi2MXiddenedbyAix;y=1ifx;y2Riand0otherwisearetheHadamardidempotentsofaBose-Mesneralgebra. ii LetfAigdi=0betheHadamardidempotentsofaBose-MesneralgebraonX.ThentherelationsRiidonXdenedbyx;y2RiifAix;y=1aretherelationsofad-classcommutativeassociationscheme. Theorem5.1.8 [5]TwoBose-Mesneralgebrasarecombinatoriallyisomorphicifandonlyiftherelatedassociationschemesareisomorphic.WenotethattheconverseofLemma5.1.2fails.Therearemanynon-isomorphicassociationschemeswhichhaveisomorphicBose-Mesneralgebras.WeshallseethatthisisthecaseformanyLatinsquares. 5.2IsomorphismsofSubconstituentAlgebrasWediscusssomenotionsofisomorphismforsubconstituentalgebras.Dependinguponhowmuchinformationabouttherelatedassociationscheme/Bose-Mesneralgebraoneretainsuponpassingtothesubconstituentalgebra,severalnotionsofisomorphismmaybedistinguished.Weusethefollowingnotation. Notation5.2.1 LetMandM0denoteBose-MesneralgebrasonXandX0,respectively.LetfAigdi=0andfA0igd0i=0denotetherespectiveHadamardidempotentsofMandM0.Fixp2Xandp02X0,andletfEigdi=0,fEi0gdi=0denotetherespectivedualidempotentsofMandM0withrespecttopandp0.LetTandT0denotetherespectivesubconstituentalgebrasofMandM0. Denition5.2.2 AdoptNotation5.2.1. 58

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i WesaythatTandT0arecombinatoriallyisomorphicwheneverthereisapermu-tationmatrixP2MX;X0andabijection:f0;1;:::;dg!f0;1;:::;d0gsuchthatAiP=PA0iandEiP=PEiid. ii WesaythatTandT0arestructurallyisomorphicwheneverthereisaninvertiblematrixHandabijection:f0;1;:::;dg!f0;1;:::;d0gsuchthatAiH=HA0iandEiH=HEiid. iii WesaythatTandT0areBose-Mesnerisomorphicwheneverthereisanalgebraisomorphism:T!T0andabijection:f0;1;:::;dg!f0;1;:::;d0gsuchthatAi=A0iandEi=Eiid. iv WesaythatTandT0areabstractlyisomorphicwheneverthereisanalgebraiso-morphismfromTtoT0.CombinatorialisomorphismforBose-MesneralgebrasnecessarilymapHadamardidem-potenttoHadamardidempotentsLemma5.1.3.Sincesubconstituentalgebrasignoreentrywisemultiplication,wemakethisassumptioninDenition5.2.2i.ClearlycombinatoriallyisomorphicimpliesstructurallyisomorphicimpliesBose-Mesnerisomorphicimpliesabstractlyisomorphic.InSection5.4weshalluseLatinsquarestoshowthatthereverseimplicationsfail.Beforewedothiswediscussthesevariousnotionsofisomorphisminfurtherdetail. Lemma5.2.3 WithNotation5.2.1,letbeabijectionfromXtoX0,letbeabijectionfromf0;1;2;:::;dgtof0;1;2;:::;d0g,andletPbethepermutationmatrixof.Thenthefollowingareequivalent. i isacombinatorialisomorphismfromMtoM0whichsatisesp=p0andisrelatedtoasinLemma5.1.3. ii P;isacombinatorialisomorphismfromTtoT0. 59

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Proof.iii:Computeforallx0,y02X0:PAiP)]TJ/F25 7.97 Tf 6.586 0 Td[(1x0;y0=Ai)]TJ/F25 7.97 Tf 6.587 0 Td[(1x0;)]TJ/F25 7.97 Tf 6.586 0 Td[(1y0=Aix;y=A0ix;y=A0ix0;y0;andPEiP)]TJ/F25 7.97 Tf 6.587 0 Td[(1x0;y0=Ei)]TJ/F25 7.97 Tf 6.587 0 Td[(1x0;)]TJ/F25 7.97 Tf 6.586 0 Td[(1y0=x;yEix;x=x;yAip;x=x;yA0ip;x=x;yEix;x=Eix;y=Eix0;y0:iii:Computeforallx,y2X:Aix;y=P)]TJ/F25 7.97 Tf 6.587 0 Td[(1AiPx;y=A0ix;yandEix;y=x;yEix;x=x;yAip;x=x;yP)]TJ/F25 7.97 Tf 6.587 0 Td[(1A0iPp;x=x;yA0ip;x=x;yA0ip0;x0==Eix0;y0=Eix;y:2WegivemoduletheoreticdescriptionsofstructuralandBose-Mesnerisomorphism.Weneedthefollowingnotionformodules. Denition5.2.4 WithNotation5.2.1,xorderingoftheHadamardanddualidem60

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potents.LetWandW0denoteT-andT0-modules,respectively.WesaythatWandW0aresimilarrelativetotheorderingoftheidempotentswheneverWandW0haverespectiveorderedbasisand0suchthat[Ai]=[A0i]0,[Ei]=[Ei0]0id,where[B]OdenotesthematrixrepresentingBwithrespecttothebasisO. Lemma5.2.5 WithNotation5.2.1,xorderingoftheHadamardanddualidempotents.LetWandW0denoteT-andT0-modules,respectively.ThenWandW0aresimilarwithrespecttothisorderingifandonlyifforanybasisofWandanybasis0ofW0,thepairs[Ai],[A0i]0and[Ei],[Ei0]0idaresimultaneouslysimilar.Proof.IfWandW0aresimilar,then0=I,whereI2M0;istheidentitymatrix,so[Ai]issimilarto[A0i]0and[Ei]issimilarto[Ei0]0.If[Ai]issimilarto[A0i]0,thenQ[Ai]=[A0i]0QforsomeinvertiblematrixQ.MakingachangeofbasiswithmatrixQ)]TJ/F25 7.97 Tf 6.586 0 Td[(1givestheequalitybetween[Ai],and[A0i]0.Likewisefor[Ei]and[Ei0]0.2Toshowthatmodulesarenotsimilar,itsucestoshowthattheAiandA0ihavedistinctspectraonthemodule. Theorem5.2.6 WithNotation5.2.1,TandT0arestructurallyisomorphicifandonlyifforsomeorderingsoftheHadamardanddualidempotentsthefollowinghold. i EveryirreducibleT-moduleissimilartosomeirreducibleT0-modulerelativetothegivenorderingsoftheidempotents. ii IfWandW0arerespectivelysimilarirreduciblemodulesofTandT0,thenmultW=multW0.Proof.Let=f1;:::sgand0=f01;:::0sgbeindexsetsfortheisomorphismclassesofirreducibleT-andT0-modules,respectively.Let'1;:::;'n,and'001;:::;'00nbethecorrespondingprimitivecentralidempotentofTandT0.Recallthat'iVisthesumofallirreducibleT-modulesintheisomorphismclassindexedbyi,andthatV=Pni=1'iV.Similarly,V0=Pmi=1'00iV0.SupposeTandT0arestructurallyisomorphic.ThenthereexistsaninvertiblematrixHandabijection:f0;1;:::;dg!f0;1;:::;d0gsuchthatAiH=HA0iandEiH= 61

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HEiid.Deneamap:V0!Vbyv0=Hv0.ThenmapsirreducibleT0-modulestosimilarirreducibleT-modules.IndeedifWisanirreducibleT-modulewithbasisfw1;w2;:::;wkg,thenEiv0=EiHv0=HEjv02HW0=W0andAiv0=AiHv0=HA0jv02HW0.ThusHW0isaT-module.AlsonotethatHwitnessesthatWandW0aresimilar.SinceHisinvertible,itrespectslinearindependence,somultW=multW.Nowsupposeiandiihold.For1is,takeadirectdecompositionof'iVintoirreducibleT-modulesWi1Wi2Wim,wherem=multi.For1jm,xabasisij=fwijkgdk=0ofWij,whered=dimWij,suchthat[A`]ij1=[A`]ij2for1j1;j2m.SaytheirreducibleT0modulesin'0iV0aresimilartothosein'iV.For1is,takeadirectdecomposition'0iV0intoirreducibleT0-modulesWi10Wi20Wi0m.Thenumberofdirectsummandsforeachiagreebyii.For1jm,xabasisij0=fwijk0gdk=0ofWij0suchthat[A`]ij=[A0`]ij0and[E`]ij=[E`]ij0forall`.DeneamatrixHbyHwijk=wijk0.ObservethatmultiplicationbyHrespectslinearindependence,soitmapsthen2-dimensionalvectorspaceVtoann2-dimensionalvectorsubspaceofV0,soitmapsVontoV0.ThusHisinvertible.OnenowchecksusingsimilaritythatA`H=HA0`andE`H=HE`.ToregaintheinitialorderingoftheHadamardidempotents,somepermutationmayberequired.ThenH;,denesastructuralisomorphism.2 Theorem5.2.7 WithNotation5.2.1,TandT0areBose-MesnerisomorphicifandonlyifforsomeorderingsoftheHadamardanddualidempotentseveryirreducibleT-moduleissimilartosomeirreducibleT0-moduleandeveryirreducibleT0-moduleissimilartosomeirreducibleT-module.Proof.AssumethatanddeneaBose-MesnerisomorphismfromTtoT0,thatis,Ai=A0iandEi=Eiforalli.LetW0beanirreducibleT0-modules.DeneaT-modulestructureonW0byAiv0=Aiv0=A0iv0andEiv0=Eiv0=Eiv0forallv02W0.Observethat[Ai]=[A0i]and[Ei]=[Ei]foranybasisofW0.ThusforeachirreducibleT0-modulethereisasimilarT-module.Asimilarargumentusing)]TJ/F25 7.97 Tf 6.587 0 Td[(1showsthatforeachirreducibleT-modulethereisasimilarT0-module. 62

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ByWedderburntheoryeach'Tisisomorphictoafullcomplexmatrixalgebra.Onesuchanisomorphismisdenedasfollows.FixanirreducibleT-moduleW'V,andxanorderedbasisofW.Lets=dimW.Dene:'T!MsbyA=[A],whereA2'T.Thenisclearlyanisomorphism.sinceT=L2'T,wemaydeneanisomoprhism=P2,whichmapsTtothedirectsumoffullcomplexmatrixalgebras.WehavesimilarmapsforT0,denotedwith0.AssumethatforeachirreducibleT0-modulethereisasimilarT-moduleandthatforeachirreducibleT-modulethereisasimilarT0-module.Thenthereisabijection:!0betweentherespectiveindexsetsfortheisomorphismclassesofirreduciblemodulesforTandT0.Bysimilarlity,takinganappropriatechoiceofbasisintheaboveconstructionwehaveAi=00A0iandEi=00Ei.Now0)]TJ/F25 7.97 Tf 6.587 0 Td[(1denesanisomorphismwiththedesiredproperties.2 Theorem5.2.8 WithNotation5.2.1,TandT0areabstractlyisomorphicifandonlyiftheyhavethesamenumberofisomorphismclassesofirreduciblemodulesofeachdimen-sion.Proof.Subconstituentalgebrasaresemisimple.ByWedderburntheory,thereisonedirectsummandinthefullmatrixdecompositionforeachisomorphismclassofirreducibleT-modules,itisddwherethemodulehasdimensiond.2 5.3EquivalencesofLatinSquaresWediscusssomenotionsofequivalenceforLatinsquares.ItturnsoutthatcombinatorialisomorphismoftheirBose-Mesneralgebrascoincideswithoneoftheseequivalences.See[42,43,69]forfurtherdiscussionoftheseequivalencerelations. Denition5.3.1 LetLbeaLatinsquareofordernwithsymbolsetf1;2;:::;ng,andletXL=fi;j;Li;jj1i;jng.NotethatLisuniquelydeterminedbyXL.TheLatinsquarepropertygivesthatuniformlypermutingthecoordinatesofthetriplesinXLgivesasetXL0forsomeLatinsquareL0.InthiscaseLandL0aresaidtobeconjugatesofoneanother. 63

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Lemma5.3.2 AnypermutationoftherelationsR1,R2,R3associatedwithaLatinsquareinducesacombinatorialautomorphismoftheassociatedBose-Mesneralgebra.Proof.Clearfromthedenitionofacombinatorialisomorphism.2 Lemma5.3.3 TheBose-MesneralgebrasofconjugateLatinsquaresarecombinatoriallyisomorphic.Proof.ImmediatefromLemma5.3.2.2 Denition5.3.4 LatinsquaresLandL0aresaidtobeisotopicwheneverL0fi;gj=hLi;jforsomerowpermutationf,columnpermutationg,andbijectionhfromthesymbolsofLtothesymbolsofL0. Lemma5.3.5 IsotopicLatinsquareshavecombinatoriallyisomorphicBose-Mesneral-gebras.Proof.SupposeLandL0areisotopicviaisotopyf;g;h.LetX=fi;j;Li;jgni;j=1andX0=ffi;gj;hLi;jgni;j=1.DenotetheassociationschemesdenedbyLandL0byX;fRig4i=0andX0;fR0ig4i=0,respectively.Dene:X!X0suchthati;j;Li;j=fi;gj;hLi;j.Clearlyisabijection.Letdenotetheidentitymaponf0;1;2;3;4g.Nowanddeneanassociationschemeisomorphism,sotheBose-MesneralgebrasofLandL0arecombinatoriallyisomorphic.2 Denition5.3.6 ThemainclassofaLatinsquareistheunionoftheisotopyclassesofitsconjugates.TwoLatinsquaresLandL0aremainclassequivalentiftheybelongtothesamemainclass,thatis,ifLisisotopictoaconjugateofL0.InTables5.1and5.2werecallsomecountsofLatinsquaresfromtheOn-lineEncy-clopediaofIntegerSequencesOEIS[82]. Theorem5.3.7 MainclassequivalentLatinsquareshavecombinatoriallyisomorphicBose-Mesneralgebras. 64

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nReducedLatinsquaresofsizenAllLatinsquaresofsizenSequenceA000315inOEISSequenceA002860inOEIS 11121231124457655616128069408812851200716942080614794199040008535281401856108776032459082956800937759757096425881655247514961568928425312256001075807214831601328114892809982437658213039871725064756920320000 Table5.1:ThenumbersofLatinsquaresofvarioussizes nMainclassesIsotopyclassesSequenceA003090inOEISSequenceA040082inOEIS 111211311422522612227147564828365716762679192708535411156187215331034817397894749939208904371354363006 Table5.2:EquivalenceclassesofLatinsquares 65

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Proof.ClearfromLemmas5.3.3and5.3.5.2 Lemma5.3.8 ThealgebraicisomorphismclassoftheBose-MesneralgebraofaLatinsquaredependsonlyuponitsorder.Proof.ImmediatefromTheorems3.1.1and5.1.4.2 5.4IsomorphismsandLatinSquares Notation5.4.1 LetLdenoteaLatinsquareofordern5andwithsymbolsetf1;2;:::;ng.LetXdenotethesetfi;j;Li;jj1i;jng.LetMdenotetheBose-MesneralgebraofL.Fixp=rp;cp;ep2X,andletTdenotethesubconstitutentalgebraofMwithrespecttorp;cp;ep.WeusethesamenotationforasecondLatinsquareL0,witha0attached. Theorem5.4.2 WithNotation5.4.1, i SupposeLandL0aremainclassequivalentviaconjugationandisotopy=f;g;h.Alsosupposep=p0.ThenTandT0arecombinatoriallyisomorphic. ii SupposeTandT0arecombinatoriallyisomorphic.ThenLandL0aremainclassequivalent.Proof.iExtendby=0,=4.Let:X!X0bethecompositionofand.Observethatisabijection.LetP2MX;X0bethematrixof.ThenP;isacombinatorialisomorphismfromTtoT0.iiLetP;beacombinatorialisomorphismfromTtoT0.Observethatxes0and4,soletbetherestrictionoftof1;2;3g,andviewaspermutingthecoordinatesofX0,thatis,aconjugationmap.WerstintroduceaLatinsquare^LconjugatetoLby.ClearlyLand^Laremainclassequivalent.Denotetheobjectsassociatedwith^Lwith^.ThusinducesacombinatorialisomorphismfromTto^TsendingA2Tto^A2^Twith^Ax;y=Ax;y.Inparticular,Aimapsto^AiandEimapsto^Ei.Deneapermutation:^X!X0by)]TJ/F25 7.97 Tf 6.587 0 Td[(1^x=Px.Weclaimthatthisinducesanisotopyof^LtoL.Say^x=^r1;^c1;^e1,^w=^r1;^c2;^e22^X.Then^x=r01;c01;e01 66

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and^w=r001;c02;e02.ButsincePispartofacombinatorialisomorphism,r01=r001.Thusthereisawell-denedbijectionffromtherowsof^LtotherowsofL0inducedby.SimilarlytherearebijectionsgandhmappingcolumnsandentriesofLtocolumnsandentriesofL0.Nowf;g;hisanisotopy.Thus^LandL0aremainclassequivalent.2 Theorem5.4.3 WithNotation5.4.1,TandT0arestructurallyisomorphicifandonlyifLandL0havethesamecyclestructureswithrespecttopandp0.Proof.Wenotethattheprimarymoduleandone-dimensionalmodulesinthefourthsubconstituenthavethesamemultiplicitiesanddimensionsforTandT0andarethecorrespondingmodulesaresimilarbyTheorems2.3.8and3.7.2.SupposeTandT0arestructurallyisomorphic.Thenthemulti-setofrootsofunityarisinginconnectionwiththeirreduciblemodulesarethesameforTandT0.Takeaprimitiverootofunitywiththehighestorder.NecessarilythereisaninterleavedcycleforbothLandL0oflengthequaltothatorder.Takeawaytheotherrootsofunitythatmustariseinconnectionwiththatcyclemodule.ArguingbyinductionshowsthatLandL0havethecyclestructureswithrespecttopandp0.SupposeLandL0havethesamecyclestructureswithrespecttopandp0.FormacorrespondencebetweeninterleavedcyclesofLandL0ofthesamelength.ObservethecorrespondingcyclemodulesaresimilarbyTheorem3.3.1,sotheirirreduciblesubmodulesareaswell.ThusTandT0arestructurallyisomorphicbyTheorem5.2.6.2InTable5.3werecallthenumberofpartitionsoftheintegersfrom1to10.Thepartitionscorrespondtopossiblecyclestructuresoftheinterleavedcycles.HencethemaximumpossiblenumberofstructuralisomorphismclassesofsubconstituentalgebrasforLatinsquaresofordernisthenumberofpartitionsofn)]TJ/F19 11.955 Tf 12.48 0 Td[(1.ComparehowsmallthesevaluesareversusthoseofTable5.2. Theorem5.4.4 WithNotation5.4.1,TandT0areBose-MesnerisomorphicifandonlyifLandL0havecyclestructureswithrespecttopandp0withthesamedivisorsofthelengthsofthecyclesignoringmultiplicity. 67

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n Partitionsofn)]TJ/F15 10.909 Tf 10.909 0 Td[(1 sequenceA000041inOEIS 2 13 24 35 56 77 118 159 2210 30 Table5.3:Numberofpossiblestructuralisomorphismclasses Proof.Theisomorphismclassofanirreduciblesubmoduleofacyclemoduleisdeter-minedbytheassociatedrootofunity.Therootsofunityarisinginacyclemoduleoflengthkhaveorderadivisorofk,andarootarisesforeverysuchdivisor.TheresultfollowsfromTheorem5.2.7.2InlightofTheorem5.4.4,subconstituentalgebrasofLatinsquareswithcyclestruc-tures42,224,1224,and144areBose-Mesnerisomorphic,asthedistinctdivisorsineachcaseare1,2,and4.Theassociatedrootsofunityare1,)]TJ/F19 11.955 Tf 9.299 0 Td[(1,i,)]TJ/F24 11.955 Tf 9.299 0 Td[(i,wherei2=)]TJ/F19 11.955 Tf 9.298 0 Td[(1.Notethat4isnotadivisorof162soarelatedsubconstituentalgebrawillnotbeBose-Mesnerisomorphictothoseonthislist. n PossibleBose-Mesner isomorphismclasses 5 46 67 78 109 1210 19 Table5.4:NumberofpossibleBose-Mesnerisomorphismclasses Theorem5.4.5 WithNotation5.4.1,TandT0areabstractlyisomorphicifandonlyifthenumberofdistinctrootsofunitywithorderadivisorofanycyclelengthslessoneifthereisjustonecycleisthesameforeachLatinsquarewithrespecttothebasepoints. 68

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Proof.ReferringtotheproofofTheorem5.2.8,intheLatinsquarecasethereisa5-dimensionalprimarymodule,therearen2)]TJ/F19 11.955 Tf 11.553 0 Td[(6n+7mutuallyisomorphism1-dimensionalmoduleandforeachcycleoflengthktherearekrootsofunity,butwhentwomodulesareconstructedwiththesameroottheyareisomorphic.2InlightofTheorem5.4.5,subconstituentalgebrasofLatinsquareswithcyclestruc-tures123and15areabstractlyisomorphicsinceeachuses5distinctrootsofunitytodenetheirreduciblemodules. Corollary5.4.6 WithNotation5.4.1,thesubconstituentalgebraT,ofaLatinsquareLwithrespecttopisisomorphicasacomplexalgebratoM5M`6M1,where`isthenumberofmutuallynonisomorphicirreducibleT-modulesofdimension6.Proof.ClearformTheorem5.4.5.2 n Possibleabstract isomorphismclasses 5 36 47 58 69 710 8 Table5.5:Numberofpossibleabstractisomorphismclasses Example5.4.7 ConsidertheLatinsquareL=0BBBBBBBB@12345241533521443521514321CCCCCCCCAwithrespecttobasepointsp=;1;1andp0=;3;3.Therespectivecyclestructuresare41and1131.ByLemmas3.4.2and3.4.4,theirreduciblemoduleswithrespecttopare 69

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associatedwithrootsofunity)]TJ/F19 11.955 Tf 9.298 0 Td[(1,i,)]TJ/F24 11.955 Tf 9.298 0 Td[(i,wherei2=)]TJ/F19 11.955 Tf 9.299 0 Td[(1andtheirreduciblemoduleswithrespecttop0areassociatedwithrootsofunity1,,2,whereisaprimitivecubedrootofunity.Ineachcasetherearethreeisomorphismclassesof6-dimensionalirreduciblemodules.Inparticular,thesubconstituentalgebrasTpandT0p0ofLareabstractlyisomorphicbyTheorem5.4.5.HowevertheyarenotBose-MesnerisomorphicbyTheorem5.4.4. Example5.4.8 ConsidertheLatinsquareL=0BBBBBBBBBBB@1234562451633562414613255146316325141CCCCCCCCCCCAwithrespecttobasepointsp=;1;2andp0=;6;4.Therespectivecyclestructuresare1122and1321.ByLemmas3.4.2and3.4.4,theirreduciblemoduleswithrespecttopareassociatedwithrootsofunity1and)]TJ/F19 11.955 Tf 9.298 0 Td[(1,andtheirreduciblemoduleswithrespecttop0arealsoassociatedwithrootsofunity1,)]TJ/F19 11.955 Tf 9.299 0 Td[(1.Intheformercase,themodulesassociatedwith1and-1haverespectivemultiplicities3and2,andinthelattercasethemodulesassociatedwith1and-1haverespectivemultiplicities4and1.ThusthesubconstituentalgebrasareBose-MesnerisomorphicbyTheorem5.4.4,butnotstructurallyisomorphicbyTheorem5.4.3. Example5.4.9 ConsidertheLatinsquaresL=0BBBBBBBBBBB@1234562315643126454651325462136543211CCCCCCCCCCCA;L0=0BBBBBBBBBBB@1234562143653561424652315316246425131CCCCCCCCCCCA 70

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withrespecttobasepointsp=;1;1andp0=;3;6.Theirrespectivecyclestructuresare1321,and1321.ThusthesubconstituentalgebrasTpandT0p0ofLandL0arestructurallyisomorphicbyTheorem5.4.3,butnotcombinatoriallyisomorphicbyTheorem5.4.2. Example5.4.10 ConsidertheLatinsquaresL=0BBBBBBBBBBB@1234567234567134567124567123567123471234561CCCCCCCCCCCA;L0=0BBBBBBBBBBB@1234567241567331724564526731564731273612551CCCCCCCCCCCA:Itturnsoutthatwithrespecttoeverybasepoint,thecyclestructureofbothis23.Inparticular,evenconsideringallcyclestructurescannotdistinguishbetweenallLatinsquares.Wenotethatthesearethesmallestexamples. 71

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[73] M.S.MacLeanandP.Terwilliger.Tautdistance-regulargraphsandthesubcon-stituentalgebra.preprint [74] D.M.Mesner,AnewfamilyofpartiallybalancedincompleteblockdesignswithsomeLatinsquaredesignproperties,Ann.Math.Statist.38967,571{581. [75] A.MunemasaandY.Watatani,Pairesorthogonalesdesous-algbresinvolutives,C.R.Acad.Sci.ParisSr.IMath.314992,329{331. [76] M.E.Muzychuk,V-ringofpermutationgroupswithinvariantmetric,Ph.D.thesisKievStateUniversity,1989inRussian. [77] A.Neumaier,Dualityincoherentcongurations,Combinatorica9989,59{67. [78] A.Pascasio,Tightdistance-regulargraphsandtheQ-polynomialproperty.GraphsCombin.17001,no.1,149{169. [79] A.Pascasio,Aninequalityonthecosinesofatightdistance-regulargraph.LinearAlgebraAppl.325001,no.1-3,147{159. [80] A.Pascasio,Tightgraphsandtheirprimitiveidempotents.J.AlgebraicCombin.10999,no.1,47{59. [81] A.A.Pascasio.OnthemultiplicitiesoftheprimitiveidempotentsofaQ-polynomialdistance-regulargraph.Europ.J.Combin.23002,10731078. [82] N.J.A.Sloane,TheOn-LineEncyclopediaofIntegerSequences07,publishedelectronicallyatwww.research.att.com/njas/sequences/. [83] K.Tanabe,TheirreduciblemodulesoftheTerwilligeralgebrasofDoobschemes.J.Alg.Combin.6997,173195. [84] P.Terwilliger,Thesubconstituentalgebraofanassociationscheme,J.AlgebraicCombin.PartI1992,363{388;PartII2993,73{103;PartIII2993,177{210. [85] M.TomiyamaandN.YamazakiThesubconstituentalgebraofastronglyregulargraph,KyushuJ.Math.48994,323{334. 78

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Appendices 80

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AppendixA:PermittedRootsofUnity cyclestructure i;multi 41 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii111 3111 1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3211 22 )]TJ/F15 10.909 Tf 8.485 0 Td[(1122 2112 )]TJ/F15 10.909 Tf 8.485 0 Td[(1113 14 14 Table6:IrreducibleT-modulesforn=5 81

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AppendixA:Continued cyclestructure i;multi 51 )]TJ/F26 5.978 Tf 11.515 4.524 Td[(5p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=5)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(13=5)]TJ/F15 10.909 Tf 8.485 0 Td[(14=51111 4111 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii11112 3121 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=31211 3112 1)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3311 2211 )]TJ/F15 10.909 Tf 8.485 0 Td[(1123 2113 )]TJ/F15 10.909 Tf 8.485 0 Td[(1114 15 15 Table7:IrreducibleT-modulesforn=6 82

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AppendixA:Continued cyclestructure i;multi 61 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(13p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(12=3)]TJ/F15 10.909 Tf 8.485 0 Td[(12=311111 5111 1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(5p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=5)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.484 0 Td[(13=5)]TJ/F15 10.909 Tf 8.485 0 Td[(14=521111 4121 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii12112 4112 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii11113 32 1)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3222 312111 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=31311 3113 1)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3411 23 )]TJ/F15 10.909 Tf 8.485 0 Td[(1133 2212 )]TJ/F15 10.909 Tf 8.485 0 Td[(1124 2114 )]TJ/F15 10.909 Tf 8.485 0 Td[(1115 16 16 Table8:IrreducibleT-modulesforn=7 83

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AppendixA:Continued cyclestructure i;multi 71 )]TJ/F26 5.978 Tf 11.515 4.523 Td[(7p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=7)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(13=7)]TJ/F15 10.909 Tf 8.485 0 Td[(14=7)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.484 0 Td[(15=7)]TJ/F15 10.909 Tf 8.485 0 Td[(16=7111111 6111 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(13p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(12=3)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3121111 5121 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.524 Td[(5p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=5)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(13=5)]TJ/F15 10.909 Tf 8.485 0 Td[(14=5121111 5112 1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(5p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=5)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.484 0 Td[(13=5)]TJ/F15 10.909 Tf 8.485 0 Td[(14=531111 4131 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3111211 412111 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii12113 4113 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii11114 3211 1)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3322 3122 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=32311 312112 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=31411 3114 1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3511 2311 )]TJ/F15 10.909 Tf 8.485 0 Td[(1134 2213 )]TJ/F15 10.909 Tf 8.485 0 Td[(1125 2115 )]TJ/F15 10.909 Tf 8.485 0 Td[(1116 17 17 Table9:IrreducibleT-modulesforn=8 84

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AppendixA:Continued cyclestructure i;multi 81 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii)]TJ/F26 5.978 Tf 11.515 4.524 Td[(4p )]TJ/F15 10.909 Tf 8.484 0 Td[(14p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.484 0 Td[(13=4)]TJ/F15 10.909 Tf 8.485 0 Td[(13=41111111 7111 1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(7p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=7)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.484 0 Td[(13=7)]TJ/F15 10.909 Tf 8.485 0 Td[(14=7)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(15=7)]TJ/F15 10.909 Tf 8.485 0 Td[(16=72111111 6121 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(13p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(12=3)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3221111 6112 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(13p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(12=3)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3131111 5131 1)]TJ/F26 5.978 Tf 11.515 4.523 Td[(5p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=5)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(13=5)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3)]TJ/F15 10.909 Tf 8.485 0 Td[(14=52111111 512111 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.523 Td[(5p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=5)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(13=5)]TJ/F15 10.909 Tf 8.485 0 Td[(14=5131111 5113 1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(5p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=5)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.484 0 Td[(13=5)]TJ/F15 10.909 Tf 8.485 0 Td[(14=541111 42 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii12222 413111 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3111311 4122 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii13113 412112 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii12114 4114 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii11115 3221 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=31322 3212 1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3422 Table10:IrreducibleT-modulesforn=9,pt.1 85

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AppendixA:Continued cyclestructure i;multi 312211 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=32411 312113 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=31511 3115 1)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3611 24 )]TJ/F15 10.909 Tf 8.485 0 Td[(1144 2312 )]TJ/F15 10.909 Tf 8.485 0 Td[(1135 2214 )]TJ/F15 10.909 Tf 8.485 0 Td[(1126 2116 )]TJ/F15 10.909 Tf 8.485 0 Td[(1117 18 18 Table11:IrreducibleT-modulesforn=9,pt.2 86

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AppendixA:Continued cycle i;multi101 )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(5p )]TJ/F15 9.963 Tf 7.749 0 Td[(15p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.748 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(12=5)]TJ/F15 9.963 Tf 7.749 0 Td[(12=5)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.748 0 Td[(13=5)]TJ/F15 9.963 Tf 7.749 0 Td[(13=5)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(14=5)]TJ/F15 9.963 Tf 7.749 0 Td[(14=5111111111 9111 1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(9p )]TJ/F15 9.963 Tf 7.748 0 Td[(1)]TJ/F15 9.963 Tf 7.748 0 Td[(12=9)]TJ/F6 4.981 Tf 10.516 4.131 Td[(3p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.749 0 Td[(14=9)]TJ/F15 9.963 Tf 7.748 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(15=9)]TJ/F15 9.963 Tf 7.749 0 Td[(12=3)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(17=9)]TJ/F15 9.963 Tf 7.749 0 Td[(18=9211111111 8121 )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F45 9.963 Tf 7.748 0 Td[(ii1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(4p )]TJ/F15 9.963 Tf 7.749 0 Td[(14p )]TJ/F15 9.963 Tf 7.748 0 Td[(1)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(13=4)]TJ/F15 9.963 Tf 7.749 0 Td[(13=421121111 8112 )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F45 9.963 Tf 7.748 0 Td[(ii1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(4p )]TJ/F15 9.963 Tf 7.749 0 Td[(14p )]TJ/F15 9.963 Tf 7.748 0 Td[(1)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(13=4)]TJ/F15 9.963 Tf 7.749 0 Td[(13=411131111 7131 1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(7p )]TJ/F15 9.963 Tf 7.748 0 Td[(1)]TJ/F15 9.963 Tf 7.748 0 Td[(12=7)]TJ/F6 4.981 Tf 10.516 4.131 Td[(3p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.748 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(13=7)]TJ/F15 9.963 Tf 7.749 0 Td[(14=7)]TJ/F15 9.963 Tf 7.749 0 Td[(12=3)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(15=7)]TJ/F15 9.963 Tf 7.749 0 Td[(16=7211111111 712111 )]TJ/F15 9.963 Tf 7.749 0 Td[(11)]TJ/F6 4.981 Tf 10.517 4.131 Td[(7p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.749 0 Td[(12=7)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(13=7)]TJ/F15 9.963 Tf 7.749 0 Td[(14=7)]TJ/F15 9.963 Tf 7.748 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(15=7)]TJ/F15 9.963 Tf 7.749 0 Td[(16=713111111 7113 1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(7p )]TJ/F15 9.963 Tf 7.748 0 Td[(1)]TJ/F15 9.963 Tf 7.748 0 Td[(12=7)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(13=7)]TJ/F15 9.963 Tf 7.749 0 Td[(14=7)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(15=7)]TJ/F15 9.963 Tf 7.749 0 Td[(16=74111111 6141 )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F45 9.963 Tf 7.748 0 Td[(ii1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(3p )]TJ/F15 9.963 Tf 7.749 0 Td[(13p )]TJ/F15 9.963 Tf 7.748 0 Td[(1)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(12=3)]TJ/F15 9.963 Tf 7.749 0 Td[(12=321121111 613111 )]TJ/F15 9.963 Tf 7.749 0 Td[(11)]TJ/F6 4.981 Tf 10.517 4.131 Td[(3p )]TJ/F15 9.963 Tf 7.749 0 Td[(13p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.748 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(12=3)]TJ/F15 9.963 Tf 7.749 0 Td[(12=3132112 6122 )]TJ/F15 9.963 Tf 7.749 0 Td[(11)]TJ/F6 4.981 Tf 10.517 4.131 Td[(3p )]TJ/F15 9.963 Tf 7.749 0 Td[(13p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.748 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(12=3)]TJ/F15 9.963 Tf 7.749 0 Td[(12=3331111 612112 )]TJ/F15 9.963 Tf 7.749 0 Td[(11)]TJ/F6 4.981 Tf 10.517 4.131 Td[(3p )]TJ/F15 9.963 Tf 7.749 0 Td[(13p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.748 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(12=3)]TJ/F15 9.963 Tf 7.749 0 Td[(12=3241111 6114 )]TJ/F15 9.963 Tf 7.749 0 Td[(11)]TJ/F6 4.981 Tf 10.517 4.131 Td[(3p )]TJ/F15 9.963 Tf 7.749 0 Td[(13p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.748 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(12=3)]TJ/F15 9.963 Tf 7.749 0 Td[(12=3151111 514111 )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F45 9.963 Tf 7.748 0 Td[(ii1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(5p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.749 0 Td[(12=5)]TJ/F15 9.963 Tf 7.748 0 Td[()]TJ/F15 9.963 Tf 7.748 0 Td[(13=5)]TJ/F15 9.963 Tf 7.748 0 Td[(14=511131111 513121 )]TJ/F15 9.963 Tf 7.749 0 Td[(11)]TJ/F6 4.981 Tf 10.517 4.131 Td[(5p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(3p )]TJ/F15 9.963 Tf 7.748 0 Td[(1)]TJ/F15 9.963 Tf 7.749 0 Td[(12=5)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.748 0 Td[(13=5)]TJ/F15 9.963 Tf 7.749 0 Td[(12=3)]TJ/F15 9.963 Tf 7.749 0 Td[(14=513111111 513112 1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(5p )]TJ/F15 9.963 Tf 7.748 0 Td[(1)]TJ/F6 4.981 Tf 10.516 4.131 Td[(3p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.749 0 Td[(12=5)]TJ/F15 9.963 Tf 7.748 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(13=5)]TJ/F15 9.963 Tf 7.749 0 Td[(12=3)]TJ/F15 9.963 Tf 7.749 0 Td[(14=54111111 512211 )]TJ/F15 9.963 Tf 7.749 0 Td[(11)]TJ/F6 4.981 Tf 10.517 4.131 Td[(5p )]TJ/F15 9.963 Tf 7.749 0 Td[(1)]TJ/F15 9.963 Tf 7.749 0 Td[(12=5)]TJ/F15 9.963 Tf 7.749 0 Td[()]TJ/F15 9.963 Tf 7.749 0 Td[(13=5)]TJ/F15 9.963 Tf 7.749 0 Td[(14=5241111 Table12:IrreducibleT-modulesforn=10,pt.1 87

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AppendixA:Continued cycle i;multi52 1)]TJ/F26 5.978 Tf 11.515 4.523 Td[(5p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=5)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(13=5)]TJ/F15 10.909 Tf 8.485 0 Td[(14=522222 512113 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.516 4.524 Td[(5p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=5)]TJ/F15 10.909 Tf 8.484 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(13=5)]TJ/F15 10.909 Tf 8.484 0 Td[(14=5151111 5115 1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(5p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=5)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 8.485 0 Td[(13=5)]TJ/F15 10.909 Tf 8.485 0 Td[(14=561111 4221 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii13223 4212 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii12224 4132 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii1)]TJ/F26 5.978 Tf 11.516 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3111322 41312111 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii1)]TJ/F26 5.978 Tf 11.516 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3211411 413113 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii1)]TJ/F26 5.978 Tf 11.516 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3111511 4123 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii14114 412212 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii13115 412114 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii12116 4116 )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F45 10.909 Tf 8.485 0 Td[(ii11117 3311 1)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3433 Table13:IrreducibleT-modulesforn=10,pt.2 88

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AppendixA:Continued cycle i;multi 3222 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=32422 322112 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=31522 3214 1)]TJ/F26 5.978 Tf 11.516 4.523 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3622 312311 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=33511 312213 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=32611 312115 )]TJ/F15 10.909 Tf 8.485 0 Td[(11)]TJ/F26 5.978 Tf 11.515 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=31711 3117 1)]TJ/F26 5.978 Tf 11.516 4.524 Td[(3p )]TJ/F15 10.909 Tf 8.485 0 Td[(1)]TJ/F15 10.909 Tf 8.485 0 Td[(12=3811 25 )]TJ/F15 10.909 Tf 8.485 0 Td[(1155 2412 )]TJ/F15 10.909 Tf 8.485 0 Td[(1146 2314 )]TJ/F15 10.909 Tf 8.485 0 Td[(1137 2216 )]TJ/F15 10.909 Tf 8.485 0 Td[(1128 2118 )]TJ/F15 10.909 Tf 8.485 0 Td[(1119 110 110 Table14:IrreducibleT-modulesforn=10,pt.3 89

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AppendixB:ComputerCodeRoutinesforLatinSquaresBose-MesnerAlgebrasO[General::"spell1";N::"meprec"]O[General::"spell1";N::"meprec"]O[General::"spell1";N::"meprec"]LatinSquaresfromsourcestreamAcompletelistofmainclassrepresentativesuptoorder8isavailablefromGordonRoyale'swebpagehttp://cs.anu.edu.au/bdm/data/latin.htmlhttp://cs.anu.edu.au/bdm/data/latin.htmlhttp://cs.anu.edu.au/bdm/data/latin.html.Thisdataispre-sentedinlesconsistingoftherepresentativesofagivensizen.Eachlineconsistsofn^2ASCIInumerals0throughn-1,therstnnumeralsgivetherstrowoftheLatinsquare,thesecondnnumeralsgivethesecondrow,etc.Weusethisdatabyrstsetting3globalvariablestodescribetheinput:LSSource=OpenRead["latin mc7.txt"];*Changelenameasrequired*LSSource=OpenRead["latin mc7.txt"];*Changelenameasrequired*LSSource=OpenRead["latin mc7.txt"];*Changelenameasrequired*LSOrder=Sqrt[StringLength[Read[LSSource,Word]]];LSOrder=Sqrt[StringLength[Read[LSSource,Word]]];LSOrder=Sqrt[StringLength[Read[LSSource,Word]]];LSSourceLength=SetStreamPosition[LSSource,Innity]/LSOrder^2+1;LSSourceLength=SetStreamPosition[LSSource,Innity]/LSOrder^2+1;LSSourceLength=SetStreamPosition[LSSource,Innity]/LSOrder^2+1;LSSourceLSSourceLSSourcegivestheinputle,LSOrderLSOrderLSOrdergivestheorderoftheLatinSquaresintheinputle,andLSSourceLengthLSSourceLengthLSSourceLengthgivesthenumberofLatinSquaresinthele.WethenuseLSFromStream[k]LSFromStream[k]LSFromStream[k]toretrievethek-thLatinSquareinthele.LSFromStream[k ]:=LSFromStream[k ]:=LSFromStream[k ]:=Module[fRawLSg;Module[fRawLSg;Module[fRawLSg;SetStreamPosition[LSSource;SetStreamPosition[LSSource;SetStreamPosition[LSSource;k)]TJ/F19 11.955 Tf 11.955 0 Td[(1LSOrder^2+1)]TJ/F19 11.955 Tf 11.955 0 Td[(If[k===1;0;1]];k)]TJ/F19 11.955 Tf 11.955 0 Td[(1LSOrder^2+1)]TJ/F19 11.955 Tf 11.956 0 Td[(If[k===1;0;1]];k)]TJ/F19 11.955 Tf 11.955 0 Td[(1LSOrder^2+1)]TJ/F19 11.955 Tf 11.955 0 Td[(If[k===1;0;1]]; 90

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AppendixB:Continued RawLS=Read[LSSource;Word];RawLS=Read[LSSource;Word];RawLS=Read[LSSource;Word];Table[Table[Table[ToExpression[StringTake[RawLS;fLSOrderi+jg]]+ToExpression[StringTake[RawLS;fLSOrderi+jg]]+ToExpression[StringTake[RawLS;fLSOrderi+jg]]+1;1;1;fi;0;LSOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1g;fj;1;LSOrderg]]fi;0;LSOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1g;fj;1;LSOrderg]]fi;0;LSOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1g;fj;1;LSOrderg]]UnlikethedataofRoyale,wenameourentires1throughn.Thisallowsustousetheentrytoindexaposition. SamplesetupSetsourceofLatinSquaresLSSource=OpenRead["./latin mc7.txt"];LSSource=OpenRead["./latin mc7.txt"];LSSource=OpenRead["./latin mc7.txt"];LSOrder=Sqrt[StringLength[Read[LSSource;Word]]];LSOrder=Sqrt[StringLength[Read[LSSource;Word]]];LSOrder=Sqrt[StringLength[Read[LSSource;Word]]];LSSourceLength=LSSourceLength=LSSourceLength=SetStreamPosition[LSSource;Innity]=SetStreamPosition[LSSource;Innity]=SetStreamPosition[LSSource;Innity]=LSOrder^2+1;LSOrder^2+1;LSOrder^2+1;SetInstancetostudySettheLatinsquaretobestudied,andthebasepointtobeconsidered.Wesetsomenewglobalvariables,thatneednotcomefromaleasdescribedabove.LSToStudy=LSFromStream[10];LSToStudy=LSFromStream[10];LSToStudy=LSFromStream[10];LSToStudyOrder=Length[LSToStudy];LSToStudyOrder=Length[LSToStudy];LSToStudyOrder=Length[LSToStudy];BaseToStudy=1;BaseToStudy=1;BaseToStudy=1;ThisdatacanbechangedduringtheruntostudyotherLatinSquaresorotherbaespoints.ThevalueoftheparameterBaseToStudyisanumberfrom1toLSToStudyOrderLSToStudyOrderLSToStudyOrder^2.Thisisexplainedimmediatelybelow. 91

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AppendixB:Continued ThesetXWeneedtoturnthetwo-dimensionaldataofpositionsofaLatinSquareintoone-dimensionalindexesforourBose-Mesneralgebraandvice-versa.ThatistosaywemustgobetweenviewingXasasetofnumbersbetween1andn^2andatriplei,j,Li,j.Weorderthetriplesi,j,Li,jlexographicaly:i,j,Li,j0;colof[nn ]:=If[Mod[nn;LSToStudyOrder]>0;colof[nn ]:=If[Mod[nn;LSToStudyOrder]>0;Mod[nn;LSToStudyOrder];Mod[nn;LSToStudyOrder];Mod[nn;LSToStudyOrder];LSToStudyOrder]LSToStudyOrder]LSToStudyOrder]entryof[pos ]:=LSToStudy[[rowof[pos];colof[pos]]];entryof[pos ]:=LSToStudy[[rowof[pos];colof[pos]]];entryof[pos ]:=LSToStudy[[rowof[pos];colof[pos]]];posof[i ;j ]:=j+LSToStudyOrderi)]TJ/F19 11.955 Tf 11.955 0 Td[(1posof[i ;j ]:=j+LSToStudyOrderi)]TJ/F19 11.955 Tf 11.956 0 Td[(1posof[i ;j ]:=j+LSToStudyOrderi)]TJ/F19 11.955 Tf 11.955 0 Td[(1Xof[pos Integer]:=Xof[pos Integer]:=Xof[pos Integer]:=frowof[pos];colof[pos];entryof[pos]gfrowof[pos];colof[pos];entryof[pos]gfrowof[pos];colof[pos];entryof[pos]gXof[pos List]:=Map[Xof;pos]Xof[pos List]:=Map[Xof;pos]Xof[pos List]:=Map[Xof;pos]DualBose-Mesneralgebra[A ;b ]:=DiagonalMatrix[A[[b]]][A ;b ]:=DiagonalMatrix[A[[b]]][A ;b ]:=DiagonalMatrix[A[[b]]]BMalgebraNowthataLatinsquareisxedwemaydenetheHadamardidempotentsoftheasso-ciatedBose-Mesneralgebra.ThesematricesneedtoberedenedwheenvertheLatin 92

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AppendixB:Continued Squaretobestudiedischanged.A[0]=IdentityMatrix[LSToStudyOrder^2];A[0]=IdentityMatrix[LSToStudyOrder^2];A[0]=IdentityMatrix[LSToStudyOrder^2];J=Table[1;fi;1;LSToStudyOrder^2g;J=Table[1;fi;1;LSToStudyOrder^2g;J=Table[1;fi;1;LSToStudyOrder^2g;fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];A[1]=A[1]=A[1]=Table[If[And[rowof[i]==rowof[j];i6=j];1;0];Table[If[And[rowof[i]==rowof[j];i6=j];1;0];Table[If[And[rowof[i]==rowof[j];i6=j];1;0];fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];A[2]=A[2]=A[2]=Table[If[And[colof[i]==colof[j];i6=j];1;0];Table[If[And[colof[i]==colof[j];i6=j];1;0];Table[If[And[colof[i]==colof[j];i6=j];1;0];fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];A[3]=A[3]=A[3]=Table[Table[Table[If[And[LSToStudy[[rowof[i];colof[i]]]If[And[LSToStudy[[rowof[i];colof[i]]]If[And[LSToStudy[[rowof[i];colof[i]]]======LSToStudy[[rowof[j];colof[j]]];LSToStudy[[rowof[j];colof[j]]];LSToStudy[[rowof[j];colof[j]]];i6=j];i6=j];i6=j];1;0];1;0];1;0];fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];A[4]=J)]TJ/F19 11.955 Tf 11.955 0 Td[(A[0]+A[1]+A[2]+A[3];A[4]=J)]TJ/F19 11.955 Tf 11.956 0 Td[(A[0]+A[1]+A[2]+A[3];A[4]=J)]TJ/F19 11.955 Tf 11.955 0 Td[(A[0]+A[1]+A[2]+A[3];dualBMalgebraNowthataLatinsquareandbasepointarexedwemaydenethedualidempotentsoftheassociatedsubconstituentalgebra.Es[0]:=[A[0];BaseToStudy];Es[0]:=[A[0];BaseToStudy];Es[0]:=[A[0];BaseToStudy];Es[1]:=[A[1];BaseToStudy];Es[1]:=[A[1];BaseToStudy];Es[1]:=[A[1];BaseToStudy]; 93

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AppendixB:Continued Es[2]:=[A[2];BaseToStudy];Es[2]:=[A[2];BaseToStudy];Es[2]:=[A[2];BaseToStudy];Es[3]:=[A[3];BaseToStudy];Es[3]:=[A[3];BaseToStudy];Es[3]:=[A[3];BaseToStudy];Es[4]:=[A[4];BaseToStudy];Es[4]:=[A[4];BaseToStudy];Es[4]:=[A[4];BaseToStudy];SRGsB1[0]=A[0];B1[1]=A[1];B1[0]=A[0];B1[1]=A[1];B1[0]=A[0];B1[1]=A[1];B1[2]=A[2]+A[3]+A[4];B1[2]=A[2]+A[3]+A[4];B1[2]=A[2]+A[3]+A[4];Fs1[0]=Es[0];Fs1[1]=Es[1];Fs1[0]=Es[0];Fs1[1]=Es[1];Fs1[0]=Es[0];Fs1[1]=Es[1];Fs1[2]=Es[2]+Es[3]+Es[4];Fs1[2]=Es[2]+Es[3]+Es[4];Fs1[2]=Es[2]+Es[3]+Es[4];B2[0]=A[0];B2[1]=A[1]+A[2];B2[0]=A[0];B2[1]=A[1]+A[2];B2[0]=A[0];B2[1]=A[1]+A[2];B3[2]=A[3]+A[4];B3[2]=A[3]+A[4];B3[2]=A[3]+A[4];Fs2[0]=Es[0];Fs2[1]=Es[1]+Es[2];Fs2[0]=Es[0];Fs2[1]=Es[1]+Es[2];Fs2[0]=Es[0];Fs2[1]=Es[1]+Es[2];Fs2[2]=Es[3]+Es[4];Fs2[2]=Es[3]+Es[4];Fs2[2]=Es[3]+Es[4];B3[0]=A[0];B3[1]=A[1]+A[2]+A[3];B3[0]=A[0];B3[1]=A[1]+A[2]+A[3];B3[0]=A[0];B3[1]=A[1]+A[2]+A[3];B3[2]=A[4];B3[2]=A[4];B3[2]=A[4];Fs3[0]=Es[0];Fs3[1]=Es[1]+Es[2]+Es[3];Fs3[0]=Es[0];Fs3[1]=Es[1]+Es[2]+Es[3];Fs3[0]=Es[0];Fs3[1]=Es[1]+Es[2]+Es[3];Fs3[2]=Es[4];Fs3[2]=Es[4];Fs3[2]=Es[4];LatinsquarestructureandthesubconstituentalgebraInterleavedcycles,cyclestructureofLatinSquaresTheroutineInterleavedCycles[L,p]InterleavedCycles[L,p]InterleavedCycles[L,p]takesasinputaLatinSquareLandabasepointp.ItreturnsthefulllistofinterleavedcyclesTheeachinterleavedcycleconsistsofthreelist{C1,C2,C3{eachwithelementsorderedasusual.InterleavedCycles[L ;p ]:=Module[InterleavedCycles[L ;p ]:=Module[InterleavedCycles[L ;p ]:=Module[fLprow=DeleteCases[fLprow=DeleteCases[fLprow=DeleteCases[Table[fp[[1]];i;L[[p[[1]];i]]g;Table[fp[[1]];i;L[[p[[1]];i]]g;Table[fp[[1]];i;L[[p[[1]];i]]g; 94

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AppendixB:Continued fi;1;Length[L[[1]]]g];fi;1;Length[L[[1]]]g];fi;1;Length[L[[1]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];*Alltriplesi;j;Li;jinthesame*Alltriplesi;j;Li;jinthesame*Alltriplesi;j;Li;jinthesamerowasp;excpetp*rowasp;excpetp*rowasp;excpetp*Lpcol=DeleteCases[Lpcol=DeleteCases[Lpcol=DeleteCases[Table[fi;p[[2]];L[[i;p[[2]]]]g;Table[fi;p[[2]];L[[i;p[[2]]]]g;Table[fi;p[[2]];L[[i;p[[2]]]]g;fi;1;Length[L[[1]]]g];fi;1;Length[L[[1]]]g];fi;1;Length[L[[1]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];*Alltriplesi;j;Li;jinthesame*Alltriplesi;j;Li;jinthesame*Alltriplesi;j;Li;jinthesamecolumnasp;excpetp*columnasp;excpetp*columnasp;excpetp*Lpent=DeleteCases[Lpent=DeleteCases[Lpent=DeleteCases[Map[Function[Append[#;L[[#[[1]];#[[2]]]]]];Map[Function[Append[#;L[[#[[1]];#[[2]]]]]];Map[Function[Append[#;L[[#[[1]];#[[2]]]]]];Position[L;Position[L;Position[L;L[[p[[1]];p[[2]]]]]];L[[p[[1]];p[[2]]]]]];L[[p[[1]];p[[2]]]]]];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];*Alltriplesi;j;Li;jwiththe*Alltriplesi;j;Li;jwiththe*Alltriplesi;j;Li;jwiththesameentryasp;excpetp*sameentryasp;excpetp*sameentryasp;excpetp*atr;atc;ate;currentrcycle;currentccycle;atr;atc;ate;currentrcycle;currentccycle;atr;atc;ate;currentrcycle;currentccycle;currentecycle;Interleavedcyclelist=fgg;currentecycle;Interleavedcyclelist=fgg;currentecycle;Interleavedcyclelist=fgg;While[Length[Lprow]>0;While[Length[Lprow]>0;While[Length[Lprow]>0;currentrcycle=fg;currentrcycle=fg;currentrcycle=fg;currentccycle=fg;currentccycle=fg;currentccycle=fg;currentecycle=fg;currentecycle=fg;currentecycle=fg;atr=Lprow[[1]];atr=Lprow[[1]];atr=Lprow[[1]];*beginwithrstrowelementnotyetused**beginwithrstrowelementnotyetused**beginwithrstrowelementnotyetused*While[Not[MemberQ[currentrcycle;atr]];While[Not[MemberQ[currentrcycle;atr]];While[Not[MemberQ[currentrcycle;atr]];currentrcycle=Append[currentrcycle;atr];currentrcycle=Append[currentrcycle;atr];currentrcycle=Append[currentrcycle;atr];atc=atc=atc= 95

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AppendixB:Continued Select[Lpcol;Function[#[[3]]===atr[[3]]]][[Select[Lpcol;Function[#[[3]]===atr[[3]]]][[Select[Lpcol;Function[#[[3]]===atr[[3]]]][[1]];1]];1]];currentccycle=Append[currentccycle;atc];currentccycle=Append[currentccycle;atc];currentccycle=Append[currentccycle;atc];ate=ate=ate=Select[Lpent;Function[#[[1]]===atc[[1]]]][[Select[Lpent;Function[#[[1]]===atc[[1]]]][[Select[Lpent;Function[#[[1]]===atc[[1]]]][[1]];1]];1]];currentecycle=Append[currentecycle;ate];currentecycle=Append[currentecycle;ate];currentecycle=Append[currentecycle;ate];atr=atr=atr=Select[Lprow;Function[#[[2]]===ate[[2]]]][[Select[Lprow;Function[#[[2]]===ate[[2]]]][[Select[Lprow;Function[#[[2]]===ate[[2]]]][[1]];];1]];];1]];];Interleavedcyclelist=Interleavedcyclelist=Interleavedcyclelist=Prepend[Interleavedcyclelist;Prepend[Interleavedcyclelist;Prepend[Interleavedcyclelist;fcurrentrcycle;fcurrentrcycle;fcurrentrcycle;currentccycle;currentecycleg];currentccycle;currentecycleg];currentccycle;currentecycleg];Lprow=Select[Lprow;Lprow=Select[Lprow;Lprow=Select[Lprow;Function[Not[MemberQ[currentrcycle;#]]]];Function[Not[MemberQ[currentrcycle;#]]]];Function[Not[MemberQ[currentrcycle;#]]]];*removetheelementsofthisrowcycle*removetheelementsofthisrowcycle*removetheelementsofthisrowcyclefromavailableelements*fromavailableelements*fromavailableelements*];];];InterleavedcyclelistInterleavedcyclelistInterleavedcyclelist*ReturnInterleavedcyclelist**ReturnInterleavedcyclelist**ReturnInterleavedcyclelist*]]]TheroutineLSCycleStructure[L,p]LSCycleStructure[L,p]LSCycleStructure[L,p]takesasinputaLatinSquareLandabasepointp.Itreturnsasortedlistconsistingofthelengthsoftheinterleavedcycles,LSCycleStructure[L ;p ]:=Module[LSCycleStructure[L ;p ]:=Module[LSCycleStructure[L ;p ]:=Module[fLprow=DeleteCases[fLprow=DeleteCases[fLprow=DeleteCases[Table[fp[[1]];i;L[[p[[1]];i]]g;Table[fp[[1]];i;L[[p[[1]];i]]g;Table[fp[[1]];i;L[[p[[1]];i]]g;fi;1;Length[L[[1]]]g];fi;1;Length[L[[1]]]g];fi;1;Length[L[[1]]]g]; 96

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AppendixB:Continued fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];Lpcol=DeleteCases[Lpcol=DeleteCases[Lpcol=DeleteCases[Table[fi;p[[2]];L[[i;p[[2]]]]g;Table[fi;p[[2]];L[[i;p[[2]]]]g;Table[fi;p[[2]];L[[i;p[[2]]]]g;fi;1;Length[L[[1]]]g];fi;1;Length[L[[1]]]g];fi;1;Length[L[[1]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];Lpent=DeleteCases[Lpent=DeleteCases[Lpent=DeleteCases[Map[Function[Append[#;L[[#[[1]];#[[2]]]]]];Map[Function[Append[#;L[[#[[1]];#[[2]]]]]];Map[Function[Append[#;L[[#[[1]];#[[2]]]]]];Position[L;L[[p[[1]];p[[2]]]]]];Position[L;L[[p[[1]];p[[2]]]]]];Position[L;L[[p[[1]];p[[2]]]]]];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];fp[[1]];p[[2]];L[[p[[1]];p[[2]]]]g];atr;atc;ate;currentcycle;cyclelen=fgg;atr;atc;ate;currentcycle;cyclelen=fgg;atr;atc;ate;currentcycle;cyclelen=fgg;*Print[Lprow];Print[Lpcol];Print[Lpent];**Print[Lprow];Print[Lpcol];Print[Lpent];**Print[Lprow];Print[Lpcol];Print[Lpent];*While[Length[Lprow]>0;While[Length[Lprow]>0;While[Length[Lprow]>0;currentcycle=fg;currentcycle=fg;currentcycle=fg;atr=Lprow[[1]];atr=Lprow[[1]];atr=Lprow[[1]];While[Not[MemberQ[currentcycle;atr]];While[Not[MemberQ[currentcycle;atr]];While[Not[MemberQ[currentcycle;atr]];currentcycle=Append[currentcycle;atr];currentcycle=Append[currentcycle;atr];currentcycle=Append[currentcycle;atr];atc=atc=atc=Select[Lpcol;Function[#[[3]]===atr[[3]]]][[Select[Lpcol;Function[#[[3]]===atr[[3]]]][[Select[Lpcol;Function[#[[3]]===atr[[3]]]][[1]];1]];1]];ate=ate=ate=Select[Lpent;Function[#[[1]]===atc[[1]]]][[Select[Lpent;Function[#[[1]]===atc[[1]]]][[Select[Lpent;Function[#[[1]]===atc[[1]]]][[1]];1]];1]];atr=atr=atr=Select[Lprow;Function[#[[2]]===ate[[2]]]][[Select[Lprow;Function[#[[2]]===ate[[2]]]][[Select[Lprow;Function[#[[2]]===ate[[2]]]][[1]];];1]];];1]];];cyclelen=Prepend[cyclelen;cyclelen=Prepend[cyclelen;cyclelen=Prepend[cyclelen;Length[currentcycle]];Length[currentcycle]];Length[currentcycle]];*Print[currentcycle];**Print[currentcycle];**Print[currentcycle];* 97

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AppendixB:Continued Lprow=Select[Lprow;Lprow=Select[Lprow;Lprow=Select[Lprow;Function[Not[MemberQ[currentcycle;#]]]];Function[Not[MemberQ[currentcycle;#]]]];Function[Not[MemberQ[currentcycle;#]]]];];];];Sort[cyclelen]Sort[cyclelen]Sort[cyclelen]]]]TheroutineAllCycleStructureInPlace[L]AllCycleStructureInPlace[L]AllCycleStructureInPlace[L]takesasinputaLatinSquareandreturnsann-by-narrayconstingofthecyclestrucuteofLwithrespecttothepositionofthecyclestrucute.AllCycleStructureInPlace[L ]:=AllCycleStructureInPlace[L ]:=AllCycleStructureInPlace[L ]:=Table[LSCycleStructure[L;fi;jg];Table[LSCycleStructure[L;fi;jg];Table[LSCycleStructure[L;fi;jg];fi;1;Length[L[[1]]]g;fj;1;Length[L[[1]]]g]fi;1;Length[L[[1]]]g;fj;1;Length[L[[1]]]g]fi;1;Length[L[[1]]]g;fj;1;Length[L[[1]]]g]ModulesReturnavectoroflengthlwitha1inpositioni.charvec[i ;l ]:=Table[If[k==i;1;0];fk;1;lg]charvec[i ;l ]:=Table[If[k==i;1;0];fk;1;lg]charvec[i ;l ]:=Table[If[k==i;1;0];fk;1;lg]PrimarymoduleReturntheprimarymoduleinaglobalvariablePrimaryModule:=PrimaryModule:=PrimaryModule:=Map[Function[#:Table[1;fi;LSToStudyOrder^2g]];Map[Function[#:Table[1;fi;LSToStudyOrder^2g]];Map[Function[#:Table[1;fi;LSToStudyOrder^2g]];Table[Es[i];fi;0;4g]]Table[Es[i];fi;0;4g]]Table[Es[i];fi;0;4g]]IntermediatemodulesThefunctionTmod[IC,div,ep]Tmod[IC,div,ep]Tmod[IC,div,ep]needsaninterleavedcycle,sayoforderk,adivisormofk,andanmthrootofunity.Wecanproducethemodulesbetweentheirreduciblesandthecyclemodules,inclusive.Weproduceacyclemodulebyusingdiv=1andep=1 98

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AppendixB:Continued Tmod[IC ;div ;ep ]:=Tmod[IC ;div ;ep ]:=Tmod[IC ;div ;ep ]:=If[And[Mod[Length[IC[[1]]];div]==0;If[And[Mod[Length[IC[[1]]];div]==0;If[And[Mod[Length[IC[[1]]];div]==0;N[ep^div]==1];N[ep^div]==1];N[ep^div]==1];Module[Module[Module[fu=fu=fu=Map[Function[fyg;Map[Function[fyg;Map[Function[fyg;Table[Apply[Plus;Table[Apply[Plus;Table[Apply[Plus;MapIndexed[Function[fx;zg;MapIndexed[Function[fx;zg;MapIndexed[Function[fx;zg;ep^z[[1]]charvec[posof[x[[1]];x[[2]]];ep^z[[1]]charvec[posof[x[[1]];x[[2]]];ep^z[[1]]charvec[posof[x[[1]];x[[2]]];LSToStudyOrder^2]];LSToStudyOrder^2]];LSToStudyOrder^2]];Take[y;foset;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;Take[y;foset;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;Take[y;foset;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;Length[IC[[1]]]=divg]]];Length[IC[[1]]]=divg]]];Length[IC[[1]]]=divg]]];foset;1;Length[IC[[1]]]=divg]];IC]g;foset;1;Length[IC[[1]]]=divg]];IC]g;foset;1;Length[IC[[1]]]=divg]];IC]g;fu[[1]];u[[2]];u[[3]];fu[[1]];u[[2]];u[[3]];fu[[1]];u[[2]];u[[3]];Map[Function[Es[4]:A[1]:Es[3]:#];u[[3]]];Map[Function[Es[4]:A[1]:Es[3]:#];u[[3]]];Map[Function[Es[4]:A[1]:Es[3]:#];u[[3]]];Map[Function[Es[4]:A[2]:Es[1]:#];u[[1]]];Map[Function[Es[4]:A[2]:Es[1]:#];u[[1]]];Map[Function[Es[4]:A[2]:Es[1]:#];u[[1]]];Map[Function[Es[4]:A[3]:Es[2]:#];u[[2]]]gMap[Function[Es[4]:A[3]:Es[2]:#];u[[2]]]gMap[Function[Es[4]:A[3]:Es[2]:#];u[[2]]]g];fg]];fg]];fg]kthroots[k ]:=x/.Solve[x^k==1]kthroots[k ]:=x/.Solve[x^k==1]kthroots[k ]:=x/.Solve[x^k==1]ProducecyclemodulesAllCycleMod:=AllCycleMod:=AllCycleMod:=Module[Module[Module[fICs=InterleavedCycles[LSToStudy;fICs=InterleavedCycles[LSToStudy;fICs=InterleavedCycles[LSToStudy;frowof[BaseToStudy];colof[BaseToStudy]g]g;frowof[BaseToStudy];colof[BaseToStudy]g]g;frowof[BaseToStudy];colof[BaseToStudy]g]g;Map[Function[Tmod[#;1;1]];ICs]]Map[Function[Tmod[#;1;1]];ICs]]Map[Function[Tmod[#;1;1]];ICs]] 99

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AppendixB:Continued ProduceirreduciblemodulesPossibleEpsilon[IC ]:=PossibleEpsilon[IC ]:=PossibleEpsilon[IC ]:=x/.Solve[x^Length[IC[[1]]]==1]x/.Solve[x^Length[IC[[1]]]==1]x/.Solve[x^Length[IC[[1]]]==1]haven'tbuiltincasethattheinterleavedcyclehaslengthn-1andepsilon=1thisispartofthetrivialmodule.IrredTmod[IC ,ep ]IrredTmod[IC ,ep ]IrredTmod[IC ,ep ]returnstheirreducibleT-moduleofacyclemoduleassociatedwithICandtherootofunityep.IrredTmod[IC ;ep ]:=IrredTmod[IC ;ep ]:=IrredTmod[IC ;ep ]:=If[ep^Length[IC[[1]]]==1;If[ep^Length[IC[[1]]]==1;If[ep^Length[IC[[1]]]==1;Module[Module[Module[fu=fu=fu=Map[Function[fyg;Map[Function[fyg;Map[Function[fyg;Apply[Plus;MapIndexed[Apply[Plus;MapIndexed[Apply[Plus;MapIndexed[Function[fx;zg;Function[fx;zg;Function[fx;zg;ep^z[[1]]charvec[posof[x[[1]];x[[2]]];ep^z[[1]]charvec[posof[x[[1]];x[[2]]];ep^z[[1]]charvec[posof[x[[1]];x[[2]]];LSToStudyOrder^2]];y]]];IC]g;LSToStudyOrder^2]];y]]];IC]g;LSToStudyOrder^2]];y]]];IC]g;fu[[1]];u[[2]];u[[3]];Es[4]:A[1]:Es[3]:u[[3]];fu[[1]];u[[2]];u[[3]];Es[4]:A[1]:Es[3]:u[[3]];fu[[1]];u[[2]];u[[3]];Es[4]:A[1]:Es[3]:u[[3]];Es[4]:A[2]:Es[1]:u[[1]];Es[4]:A[2]:Es[1]:u[[1]];Es[4]:A[2]:Es[1]:u[[1]];Es[4]:A[3]:Es[2]:u[[2]]g)]TJ/F19 11.955 Tf -106.019 0.332 Td[(Es[4]:A[3]:Es[2]:u[[2]]g)]TJ/F19 11.955 Tf -106.02 -0.332 Td[(Es[4]:A[3]:Es[2]:u[[2]]g)]TJ/F19 11.955 Tf -106.816 -20.921 Td[(KroneckerDelta[ep;1]KroneckerDelta[ep;1]KroneckerDelta[ep;1]Length[IC[[1]]]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1Length[IC[[1]]]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1Length[IC[[1]]]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1fEs[1]:J[[1]];Es[2]:J[[1]];Es[3]:J[[1]];fEs[1]:J[[1]];Es[2]:J[[1]];Es[3]:J[[1]];fEs[1]:J[[1]];Es[2]:J[[1]];Es[3]:J[[1]];Es[4]:J[[1]];Es[4]:J[[1]];Es[4]:J[[1]]gEs[4]:J[[1]];Es[4]:J[[1]];Es[4]:J[[1]]gEs[4]:J[[1]];Es[4]:J[[1]];Es[4]:J[[1]]g];fg]];fg]];fg]returnaglobalvariablewithall6-dimensionalirreducibleT-modules.All6dimIrredTmod:=All6dimIrredTmod:=All6dimIrredTmod:=Module[Module[Module[ 100

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AppendixB:Continued fICs=InterleavedCycles[LSToStudy;fICs=InterleavedCycles[LSToStudy;fICs=InterleavedCycles[LSToStudy;frowof[BaseToStudy];colof[BaseToStudy]g]g;frowof[BaseToStudy];colof[BaseToStudy]g]g;frowof[BaseToStudy];colof[BaseToStudy]g]g;Flatten[Flatten[Flatten[MapThread[Function[fIC;epvaluesg;MapThread[Function[fIC;epvaluesg;MapThread[Function[fIC;epvaluesg;Map[Function[IrredTmod[IC;#]];epvalues]];Map[Function[IrredTmod[IC;#]];epvalues]];Map[Function[IrredTmod[IC;#]];epvalues]];fICs;fICs;fICs;Map[PossibleEpsilon;ICs]g];1]]Map[PossibleEpsilon;ICs]g];1]]Map[PossibleEpsilon;ICs]g];1]]TmodParam[irred ]:=TmodParam[irred ]:=TmodParam[irred ]:=ep/.ep/.ep/.Solve[Solve[Solve[irred[[1]]:Es[1]:A[2]:Es[3]:A[1]:Es[2]:irred[[1]]:Es[1]:A[2]:Es[3]:A[1]:Es[2]:irred[[1]]:Es[1]:A[2]:Es[3]:A[1]:Es[2]:A[3]:Es[1]==epirred[[1]];ep][[1]]A[3]:Es[1]==epirred[[1]];ep][[1]]A[3]:Es[1]==epirred[[1]];ep][[1]]Fourthsubconstituent'sone-dimensionalmodulesWeproduceasetofvectorsorthogonaltoallofthecyclemodulesandtheprimarymodule.WeusetheGramSchmidtprocedurefromtheMathematicapackageLinearAlgebraOrthogonalization.RecallthatGramSchmidtpreservesthespanoftherst$i$vectorforall$i$,anddependentvectorsbecomezero.<
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AppendixB:Continued Select[Select[Select[Take[GramSchmidt[Join[m4;b4];Take[GramSchmidt[Join[m4;b4];Take[GramSchmidt[Join[m4;b4];InnerProduct!Conjugate[#1]:#2&];InnerProduct!Conjugate[#1]:#2&];InnerProduct!Conjugate[#1]:#2&];fLength[m4]+1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1g];NonZeroVecQ]]fLength[m4]+1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1g];NonZeroVecQ]]fLength[m4]+1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1g];NonZeroVecQ]]ifwealreadycomputedall6dimirredmodules,thereisnoneedtodosoagain.fourth2[SD ]:=Module[fPM=PrimaryModule;b4;m4g;fourth2[SD ]:=Module[fPM=PrimaryModule;b4;m4g;fourth2[SD ]:=Module[fPM=PrimaryModule;b4;m4g;b4=Map[Function[charvec[#;LSToStudyOrder^2]];b4=Map[Function[charvec[#;LSToStudyOrder^2]];b4=Map[Function[charvec[#;LSToStudyOrder^2]];Flatten[Position[PrimaryModule[[5]];1]]];Flatten[Position[PrimaryModule[[5]];1]]];Flatten[Position[PrimaryModule[[5]];1]]];m4=Prepend[Flatten[Map[Function[Take[#;)]TJ/F19 11.955 Tf 9.299 0 Td[(3]];SD];m4=Prepend[Flatten[Map[Function[Take[#;)]TJ/F19 11.955 Tf 9.298 0 Td[(3]];SD];m4=Prepend[Flatten[Map[Function[Take[#;)]TJ/F19 11.955 Tf 9.298 0 Td[(3]];SD];1];PM[[5]]];1];PM[[5]]];1];PM[[5]]];Select[Select[Select[Take[GramSchmidt[Join[m4;b4];Take[GramSchmidt[Join[m4;b4];Take[GramSchmidt[Join[m4;b4];InnerProduct!Conjugate[#1]:#2&];InnerProduct!Conjugate[#1]:#2&];InnerProduct!Conjugate[#1]:#2&];fLength[m4]+1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1g];NonZeroVecQ]]fLength[m4]+1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1g];NonZeroVecQ]]fLength[m4]+1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1g];NonZeroVecQ]]IrredT-moduleintoirreduciblemodulesforSRGsTakeirreducibleT-modulesasinputandreturntheirreducibleS-modulesintowhichitdecomposes.SRG3mod[im ]:=Module[fep=TmodParam[im]g;SRG3mod[im ]:=Module[fep=TmodParam[im]g;SRG3mod[im ]:=Module[fep=TmodParam[im]g;Which[Which[Which[ep==1;ep==1;ep==1;ffim[[1]]+im[[2]]+im[[3]];ffim[[1]]+im[[2]]+im[[3]];ffim[[1]]+im[[2]]+im[[3]];im[[4]]+im[[5]]+im[[6]]g;im[[4]]+im[[5]]+im[[6]]g;im[[4]]+im[[5]]+im[[6]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[2]]+im[[3]];fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[2]]+im[[3]];fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[2]]+im[[3]];im[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[5]]+im[[6]]g;im[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[5]]+im[[6]]g;im[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[5]]+im[[6]]g;fim[[1]]+2im[[2]])]TJ/F19 11.955 Tf 11.955 0 Td[(3im[[3]];fim[[1]]+2im[[2]])]TJ/F19 11.955 Tf 11.955 0 Td[(3im[[3]];fim[[1]]+2im[[2]])]TJ/F19 11.955 Tf 11.955 0 Td[(3im[[3]];im[[4]]+2im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(3im[[6]]gg;im[[4]]+2im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(3im[[6]]gg;im[[4]]+2im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(3im[[6]]gg;ep==)]TJ/F19 11.955 Tf 9.299 0 Td[(1;ep==)]TJ/F19 11.955 Tf 9.298 0 Td[(1;ep==)]TJ/F19 11.955 Tf 9.299 0 Td[(1; 102

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AppendixB:Continued ffim[[1]]+2im[[2]]+im[[3]];im[[4]]+im[[6]]g;ffim[[1]]+2im[[2]]+im[[3]];im[[4]]+im[[6]]g;ffim[[1]]+2im[[2]]+im[[3]];im[[4]]+im[[6]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[3]];im[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[5]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]])]TJ/F19 11.955 Tf 11.956 0 Td[(2im[[3]];im[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[5]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[3]];im[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[5]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]]+im[[3]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]]+im[[3]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]]+im[[3]]g;fim[[4]]+im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[6]]gg;fim[[4]]+im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[6]]gg;fim[[4]]+im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[6]]gg;True;True;True;Module[Module[Module[fthlist=fthlist=fthlist=/././.Solve[^3+3^2)]TJ/F19 11.955 Tf 11.955 0 Td[(21+Re[ep]==0];Solve[^3+3^2)]TJ/F19 11.955 Tf 11.955 0 Td[(21+Re[ep]==0];Solve[^3+3^2)]TJ/F19 11.955 Tf 11.956 0 Td[(21+Re[ep]==0];bcg;bcg;bcg;bc=bc=bc=Map[f+ep+#=#^2+2#;Map[f+ep+#=#^2+2#;Map[f+ep+#=#^2+2#;+ep+ep#=#^2+2#g&;thlist];+ep+ep#=#^2+2#g&;thlist];+ep+ep#=#^2+2#g&;thlist];Map[fim[[1]]+#[[1]]im[[2]]+#[[2]]im[[3]];Map[fim[[1]]+#[[1]]im[[2]]+#[[2]]im[[3]];Map[fim[[1]]+#[[1]]im[[2]]+#[[2]]im[[3]];#[[1]]+#[[2]]im[[4]]+#[[1]]+#[[2]]im[[4]]+#[[1]]+#[[2]]im[[4]]++#[[2]]=epim[[5]]+1+#[[1]]im[[6]]g&;+#[[2]]=epim[[5]]++#[[1]]im[[6]]g&;+#[[2]]=epim[[5]]+1+#[[1]]im[[6]]g&;bc]]bc]]bc]]]]]]]]SRG2mod[im ]:=Module[fep=TmodParam[im]g;SRG2mod[im ]:=Module[fep=TmodParam[im]g;SRG2mod[im ]:=Module[fep=TmodParam[im]g;Which[Which[Which[ep==1;ep==1;ep==1;ffim[[1]]+im[[2]];im[[4]]+im[[5]]+2im[[3]]g;ffim[[1]]+im[[2]];im[[4]]+im[[5]]+2im[[3]]g;ffim[[1]]+im[[2]];im[[4]]+im[[5]]+2im[[3]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]];im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[4]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]];im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[4]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]];im[[5]])]TJ/F19 11.955 Tf 11.956 0 Td[(im[[4]]g;fim[[3]]+im[[6]]g;fim[[3]]+im[[6]]g;fim[[3]]+im[[6]]g;fim[[4]]+im[[5]]+im[[6]])]TJ -130.146 0.332 Td[(fim[[4]]+im[[5]]+im[[6]])]TJ -130.147 -0.332 Td[(fim[[4]]+im[[5]]+im[[6]])]TJ/F19 11.955 Tf -130.944 -20.921 Td[(LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(4im[[3]]gg;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(4im[[3]]gg;LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(4im[[3]]gg;ep==)]TJ/F19 11.955 Tf 9.299 0 Td[(1;ep==)]TJ/F19 11.955 Tf 9.298 0 Td[(1;ep==)]TJ/F19 11.955 Tf 9.299 0 Td[(1;ffim[[1]]+im[[2]];im[[4]]+im[[5]]g;ffim[[1]]+im[[2]];im[[4]]+im[[5]]g;ffim[[1]]+im[[2]];im[[4]]+im[[5]]g; 103

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AppendixB:Continued fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]];im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[3]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]];im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[3]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]];im[[5]])]TJ/F19 11.955 Tf 11.956 0 Td[(im[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[3]]g;fim[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[3]]+LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3im[[3]]g;fim[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[3]]+LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3im[[3]]g;fim[[4]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[3]]+LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(3im[[3]]g;fim[[4]]+im[[5]]+im[[6]]+fim[[4]]+im[[5]]+im[[6]]+fim[[4]]+im[[5]]+im[[6]]+LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[6]]gg;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[6]]gg;LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(1im[[6]]gg;True;True;True;ffim[[1]]+im[[2]];ffim[[1]]+im[[2]];ffim[[1]]+im[[2]];im[[4]]+im[[5]]++epim[[3]]g;im[[4]]+im[[5]]++epim[[3]]g;im[[4]]+im[[5]]++epim[[3]]g;fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]];fim[[1]])]TJ/F19 11.955 Tf 11.956 0 Td[(im[[2]];fim[[1]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[2]];im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[4]]+ep)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[3]]g;im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[4]]+ep)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[3]]g;im[[5]])]TJ/F19 11.955 Tf 11.955 0 Td[(im[[4]]+ep)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[3]]g;f+epim[[3]]+im[[4]]+im[[5]]+f+epim[[3]]+im[[4]]+im[[5]]+f+epim[[3]]+im[[4]]+im[[5]]+LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[6]]g;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[6]]g;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[6]]g;f+ep^2+2epLSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[3]]+f+ep^2+2epLSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[3]]+f+ep^2+2epLSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2im[[3]]+)]TJ/F19 11.955 Tf 11.955 0 Td[(epim[[4]]+ep)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[5]]+)]TJ/F19 11.955 Tf 11.955 0 Td[(epim[[4]]+ep)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[5]]+)]TJ/F19 11.955 Tf 11.955 0 Td[(epim[[4]]+ep)]TJ/F19 11.955 Tf 11.956 0 Td[(1im[[5]]++epLSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[6]]gg+epLSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[6]]gg+epLSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(1im[[6]]gg]]]]]]SRG1mod[im ]:=Module[fep=TmodParam[im]g;SRG1mod[im ]:=Module[fep=TmodParam[im]g;SRG1mod[im ]:=Module[fep=TmodParam[im]g;ffim[[1]]g;fim[[2]]+im[[3]]+im[[4]]g;ffim[[1]]g;fim[[2]]+im[[3]]+im[[4]]g;ffim[[1]]g;fim[[2]]+im[[3]]+im[[4]]g;fim[[2]]+im[[6]]g;fim[[2]]+im[[6]]g;fim[[2]]+im[[6]]g;fepim[[2]]+im[[5]]g;fepim[[2]]+im[[5]]g;fepim[[2]]+im[[5]]g;f)]TJ/F19 11.955 Tf 11.955 0 Td[(LSToStudyOrderim[[2]]+im[[4]]g;f)]TJ/F19 11.955 Tf 11.955 0 Td[(LSToStudyOrderim[[2]]+im[[4]]g;f)]TJ/F19 11.955 Tf 11.955 0 Td[(LSToStudyOrderim[[2]]+im[[4]]g;f)]TJ/F19 11.955 Tf 15.276 0 Td[(im[[2]]+im[[3]]gg]f)]TJ/F19 11.955 Tf 15.276 0 Td[(im[[2]]+im[[3]]gg]f)]TJ/F19 11.955 Tf 15.276 0 Td[(im[[2]]+im[[3]]gg]Therountinematrep[AM,md]matrep[AM,md]matrep[AM,md]returnsthematrixrepresentingAMwithrespecttoabasismd.matrep[AM ;mbas ]:=matrep[AM ;mbas ]:=matrep[AM ;mbas ]:=Module[fvars=Table[x[i];fi;1;Length[mbas]g]g;Module[fvars=Table[x[i];fi;1;Length[mbas]g]g;Module[fvars=Table[x[i];fi;1;Length[mbas]g]g;MatrixForm[MatrixForm[MatrixForm[Flatten[Flatten[Flatten[ 104

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AppendixB:Continued vars/.Map[Solve[#==vars:mbas]&;mbas:AM];1]]]vars/.Map[Solve[#==vars:mbas]&;mbas:AM];1]]]vars/.Map[Solve[#==vars:mbas]&;mbas:AM];1]]]theroutineurep[th]returnsamatrixrepresentingamoduleoftypeUthgiveneigenvaluesr,softheadjacentymatrixurep[th ;r ;s ]:=urep[th ;r ;s ]:=urep[th ;r ;s ]:=ffth;r)]TJ/F19 11.955 Tf 11.955 0 Td[(thg;fth)]TJ/F19 11.955 Tf 11.955 0 Td[(s;r+s)]TJ/F19 11.955 Tf 11.955 0 Td[(thggffth;r)]TJ/F19 11.955 Tf 11.955 0 Td[(thg;fth)]TJ/F19 11.955 Tf 11.955 0 Td[(s;r+s)]TJ/F19 11.955 Tf 11.955 0 Td[(thggffth;r)]TJ/F19 11.955 Tf 11.955 0 Td[(thg;fth)]TJ/F19 11.955 Tf 11.955 0 Td[(s;r+s)]TJ/F19 11.955 Tf 11.956 0 Td[(thggThebasestheSRGmodulesdierfromthoseinthe"typeU"descrptionbyscalarmultiples.Notethatthediagonalentriesareinvariantandtheproductoftheo-diagonalsareinvariant.SamplerunSetInstancetostudyLSToStudy=LSFromStream[64];LSToStudy=LSFromStream[64];LSToStudy=LSFromStream[64];LSToStudyOrder=Length[LSToStudy];LSToStudyOrder=Length[LSToStudy];LSToStudyOrder=Length[LSToStudy];BaseToStudy=1;BaseToStudy=1;BaseToStudy=1;MatrixForm[LSToStudy]MatrixForm[LSToStudy]MatrixForm[LSToStudy]0BBBBBBBBBBBBBB@12345672513674312574643671255746213647135276524311CCCCCCCCCCCCCCALSCycleStructure[LSToStudy;f1;1g]LSCycleStructure[LSToStudy;f1;1g]LSCycleStructure[LSToStudy;f1;1g]f1;2;3gLSCycleStructure[LSToStudy;f1;2g]LSCycleStructure[LSToStudy;f1;2g]LSCycleStructure[LSToStudy;f1;2g]f6gLSCycleStructure[LSToStudy;f7;7g]LSCycleStructure[LSToStudy;f7;7g]LSCycleStructure[LSToStudy;f7;7g]f1;1;1;1;2g 105

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AppendixB:Continued AllCycleStructureInPlace[LSToStudy]AllCycleStructureInPlace[LSToStudy]AllCycleStructureInPlace[LSToStudy]fff1;2;3g;f6g;f6g;f6g;f6g;f2;4g;f1;2;3gg;ff6g;f6g;f1;5g;f3;3g;f1;2;3g;f2;2;2g;f2;4gg;ff6g;f1;5g;f1;2;3g;f1;5g;f6g;f6g;f1;2;3gg;ff2;4g;f2;2;2g;f1;2;3g;f6g;f1;1;2;2g;f1;1;4g;f1;2;3gg;ff6g;f1;5g;f1;5g;f6g;f1;1;4g;f1;1;2;2g;f1;1;4gg;ff2;2;2g;f1;2;3g;f1;2;3g;f1;2;3g;f1;5g;f1;5g;f1;1;1;3gg;ff1;2;3g;f1;5g;f1;2;3g;f1;1;2;2g;f2;4g;f1;5g;f1;1;1;1;2gggNotethatinthisexample,everypossiblecyclestructureexceptforonethatwithallonecyclesoccuresforsomebasepoint.LSToStudy=LSFromStream[39];LSToStudy=LSFromStream[39];LSToStudy=LSFromStream[39];LSToStudyOrder=Length[LSToStudy];LSToStudyOrder=Length[LSToStudy];LSToStudyOrder=Length[LSToStudy];BaseToStudy=1;BaseToStudy=1;BaseToStudy=1;MatrixForm[LSToStudy]MatrixForm[LSToStudy]MatrixForm[LSToStudy]0BBBBBBBBBBBBBB@12345672416753376512443516725672341612743575432161CCCCCCCCCCCCCCALSCycleStructure[LSToStudy;f7;5g]LSCycleStructure[LSToStudy;f7;5g]LSCycleStructure[LSToStudy;f7;5g]f1;1;1;1;1;1gThecyclestructuremissingfromthepreviousexampledoesoccurinthenextexample.LSToStudy=LSFromStream[38];LSToStudy=LSFromStream[38];LSToStudy=LSFromStream[38];LSToStudyOrder=Length[LSToStudy];LSToStudyOrder=Length[LSToStudy];LSToStudyOrder=Length[LSToStudy];BaseToStudy=1;BaseToStudy=1;BaseToStudy=1; 106

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AppendixB:Continued MatrixForm[LSToStudy]MatrixForm[LSToStudy]MatrixForm[LSToStudy]0BBBBBBBBBBBBBB@12345672415673317245645267315647312675312473612451CCCCCCCCCCCCCCAAllCycleStructureInPlace[LSToStudy]AllCycleStructureInPlace[LSToStudy]AllCycleStructureInPlace[LSToStudy]fff2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2gg;ff2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2gg;ff2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2gg;ff2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2gg;ff2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2gg;ff2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2gg;ff2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2g;f2;2;2gggInthisexamplethecyclestructureisthesamewithrespecttoeverybasepoint.Itismainclassequivalenttotheintegersmod7.WehavedetectednoconnectionbetweencyclestructurewithrespecttodierentbasepointsofthesameLatinsquareingeneral.Letuspickaninterestingexampleandconsideritssubconstituentalgebra.LSToStudy=LSFromStream[64];LSToStudy=LSFromStream[64];LSToStudy=LSFromStream[64];LSToStudyOrder=Length[LSToStudy];LSToStudyOrder=Length[LSToStudy];LSToStudyOrder=Length[LSToStudy];BaseToStudy=1;BaseToStudy=1;BaseToStudy=1; 107

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AppendixB:Continued BMalgebraA[0]=IdentityMatrix[LSToStudyOrder^2];A[0]=IdentityMatrix[LSToStudyOrder^2];A[0]=IdentityMatrix[LSToStudyOrder^2];J=Table[1;fi;1;LSToStudyOrder^2g;J=Table[1;fi;1;LSToStudyOrder^2g;J=Table[1;fi;1;LSToStudyOrder^2g;fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];A[1]=A[1]=A[1]=Table[If[And[rowof[i]==rowof[j];i6=j];1;0];Table[If[And[rowof[i]==rowof[j];i6=j];1;0];Table[If[And[rowof[i]==rowof[j];i6=j];1;0];fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];A[2]=A[2]=A[2]=Table[If[And[colof[i]==colof[j];i6=j];1;0];Table[If[And[colof[i]==colof[j];i6=j];1;0];Table[If[And[colof[i]==colof[j];i6=j];1;0];fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];A[3]=A[3]=A[3]=Table[Table[Table[If[And[LSToStudy[[rowof[i];colof[i]]]If[And[LSToStudy[[rowof[i];colof[i]]]If[And[LSToStudy[[rowof[i];colof[i]]]======LSToStudy[[rowof[j];colof[j]]];LSToStudy[[rowof[j];colof[j]]];LSToStudy[[rowof[j];colof[j]]];i6=j];i6=j];i6=j];1;0];1;0];1;0];fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fi;1;LSToStudyOrder^2g;fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];fj;1;LSToStudyOrder^2g];A[4]=J)]TJ/F19 11.955 Tf 11.955 0 Td[(A[0]+A[1]+A[2]+A[3];A[4]=J)]TJ/F19 11.955 Tf 11.956 0 Td[(A[0]+A[1]+A[2]+A[3];A[4]=J)]TJ/F19 11.955 Tf 11.955 0 Td[(A[0]+A[1]+A[2]+A[3];dualBMalgebraEs[0]:=[A[0];BaseToStudy];Es[0]:=[A[0];BaseToStudy];Es[0]:=[A[0];BaseToStudy];Es[1]:=[A[1];BaseToStudy];Es[1]:=[A[1];BaseToStudy];Es[1]:=[A[1];BaseToStudy];Es[2]:=[A[2];BaseToStudy];Es[2]:=[A[2];BaseToStudy];Es[2]:=[A[2];BaseToStudy]; 108

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AppendixB:Continued Es[3]:=[A[3];BaseToStudy];Es[3]:=[A[3];BaseToStudy];Es[3]:=[A[3];BaseToStudy];Es[4]:=[A[4];BaseToStudy];Es[4]:=[A[4];BaseToStudy];Es[4]:=[A[4];BaseToStudy];Sincevectorsarelistedatrows,thesematricesactontheright.Thiscontrastswiththepaperwherethevectorsarecolumnsandthematricesactontheleft.Thusthedierenceisjustatransposition.StudyexamplesICs=InterleavedCycles[LSToStudy;ICs=InterleavedCycles[LSToStudy;ICs=InterleavedCycles[LSToStudy;frowof[BaseToStudy];colof[BaseToStudy]g]frowof[BaseToStudy];colof[BaseToStudy]g]frowof[BaseToStudy];colof[BaseToStudy]g]ffff1;7;7gg;ff7;1;7gg;ff7;7;1ggg;fff1;4;4g;f1;5;5g;f1;6;6gg;ff4;1;4g;f5;1;5g;f6;1;6gg;ff4;5;1g;f5;6;1g;f6;4;1ggg;fff1;2;2g;f1;3;3gg;ff2;1;2g;f3;1;3gg;ff2;3;1g;f3;2;1ggggTherstelementisa1-cycleICs[[1]]ICs[[1]]ICs[[1]]fff1;7;7gg;ff7;1;7gg;ff7;7;1gggThesecondisa3-cycleICs[[2]]ICs[[2]]ICs[[2]]fff1;4;4g;f1;5;5g;f1;6;6gg;ff4;1;4g;f5;1;5g;f6;1;6gg;ff4;5;1g;f5;6;1g;f6;4;1gggthethirdisa2-cycle.ICs[[3]]ICs[[3]]ICs[[3]]fff1;2;2g;f1;3;3gg;ff2;1;2g;f3;1;3gg;ff2;3;1g;f3;2;1gggSomeofthecyclemodules:Tmod[ICs[[1]];1;1]Tmod[ICs[[1]];1;1]Tmod[ICs[[1]];1;1] 109

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AppendixB:Continued fff0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0gg;ff0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0gg;ff0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;1gg;ff0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;1;1;1;1;1;0gg;ff0;0;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;0gg;ff0;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;1;0;0;0;0;0;1;0;0;0;0;1;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0gggTmod[ICs[[3]];1;1]Tmod[ICs[[3]];1;1]Tmod[ICs[[3]];1;1]fff0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0g;f0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0gg;ff0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0g; 110

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AppendixB:Continued f0;0;0;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0gg;ff0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0g;f0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0gg;ff0;0;0;0;0;0;0;0;1;0;1;1;1;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0g;f0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;1;1;1;1;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0gg;ff0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0g;f0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0gg;ff0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;1;0;0;0;0;0;0;0;0;1;0;0;0;1;0;0;0g;f0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;1;0;0;0;0;0;0;0;1;0gggDatagivenaslistof6lists:therstthreelistsarethecharacteristsvectorsofthe 111

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AppendixB:Continued elementsofthecyclesinthecorrespondingsubconstituents.Thelastthreearethecharacteristicvectorsofthoseinthefourthsubconstiutents.Weturntoirreduciblemodulesc2=ICs[[2]];*a3)]TJ/F19 11.955 Tf 11.956 0 Td[(cycle*c2=ICs[[2]];*a3)]TJ/F19 11.955 Tf 11.955 0 Td[(cycle*c2=ICs[[2]];*a3)]TJ/F19 11.955 Tf 11.955 0 Td[(cycle*PossibleEpsilon[c2]PossibleEpsilon[c2]PossibleEpsilon[c2]1;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=32IrredTmod[c2;1]2IrredTmod[c2;1]2IrredTmod[c2;1]ff0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1;1;1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0g;f0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;1;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;0;0;0;0;0g;f0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;1;0;0;0;0;1;0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1g;f0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;0;0;)]TJ/F19 11.955 Tf 11.955 0 Td[(1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;1;1;1;0;1;1;0;1;1;1;1;0;1;0;1;1;0;1;1;1;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0g;f0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;1;1;1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1;1;1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1;0;1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1;1;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;1;1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1;1;1;0g;f0;0;0;0;0;0;0;0;1;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1;1;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;1;1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;0;1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0;1;1;)]TJ/F19 11.955 Tf 9.299 0 Td[(1;1;)]TJ/F19 11.955 Tf 9.298 0 Td[(1;0ggTmodParam[%]TmodParam[%]TmodParam[%]1IrredTmodc2;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3IrredTmodc2;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3IrredTmodc2;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3 112

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AppendixB:Continued 0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0g;f0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0g;f0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;f0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;0;1;1;0;1;1;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;1;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;1;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;0;1;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;1;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;1;0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;0;0;1;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;1;0;0;1;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;1;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;0;0;1;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;0;0TmodParam[%]TmodParam[%]TmodParam[%])]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3IrredTmodc2;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3IrredTmodc2;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3IrredTmodc2;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=30;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0g;f0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0; 113

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AppendixB:Continued 0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0g;f0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;f0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;0;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;0;1;1;0;1;1;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;1;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;1;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;0;1;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;1;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;1;0;0;0;0;0;0;0;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(11=3;0;0;1;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;1;0;0;1;0;0;0;)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;0;0;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;1;0;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;0;0;0;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;0;0;1;)]TJ/F19 11.955 Tf 9.299 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(11=3;0;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;0;0TmodParam[%]TmodParam[%]TmodParam[%])]TJ/F19 11.955 Tf 9.299 0 Td[(12=3SRGsB1[0]=A[0];B1[1]=A[1];B1[0]=A[0];B1[1]=A[1];B1[0]=A[0];B1[1]=A[1];B1[2]=A[2]+A[3]+A[4];B1[2]=A[2]+A[3]+A[4];B1[2]=A[2]+A[3]+A[4];Fs1[0]=Es[0];Fs1[1]=Es[1];Fs1[0]=Es[0];Fs1[1]=Es[1];Fs1[0]=Es[0];Fs1[1]=Es[1];Fs1[2]=Es[2]+Es[3]+Es[4];Fs1[2]=Es[2]+Es[3]+Es[4];Fs1[2]=Es[2]+Es[3]+Es[4];B2[0]=A[0];B2[1]=A[1]+A[2];B2[0]=A[0];B2[1]=A[1]+A[2];B2[0]=A[0];B2[1]=A[1]+A[2];B2[2]=A[3]+A[4];B2[2]=A[3]+A[4];B2[2]=A[3]+A[4];Fs2[0]=Es[0];Fs2[1]=Es[1]+Es[2];Fs2[0]=Es[0];Fs2[1]=Es[1]+Es[2];Fs2[0]=Es[0];Fs2[1]=Es[1]+Es[2]; 114

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AppendixB:Continued Fs2[2]=Es[3]+Es[4];Fs2[2]=Es[3]+Es[4];Fs2[2]=Es[3]+Es[4];B3[0]=A[0];B3[1]=A[1]+A[2]+A[3];B3[0]=A[0];B3[1]=A[1]+A[2]+A[3];B3[0]=A[0];B3[1]=A[1]+A[2]+A[3];B3[2]=A[4];B3[2]=A[4];B3[2]=A[4];Fs3[0]=Es[0];Fs3[1]=Es[1]+Es[2]+Es[3];Fs3[0]=Es[0];Fs3[1]=Es[1]+Es[2]+Es[3];Fs3[0]=Es[0];Fs3[1]=Es[1]+Es[2]+Es[3];Fs3[2]=Es[4];Fs3[2]=Es[4];Fs3[2]=Es[4];StudySRG'sS3M1=Simplify[SRG3mod[IrredTmod[c2;1]]];S3M1=Simplify[SRG3mod[IrredTmod[c2;1]]];S3M1=Simplify[SRG3mod[IrredTmod[c2;1]]];ThebasestheSRGmodulesdierfromthoseinthe"typeU"descrptionbyscalarmultiples.Notethatthediagonalentriesareinvariantandtheproductoftheo-diagonalsareinvariant.matrep[B3[1];S3M1[[1]]]matrep[B3[1];S3M1[[1]]]matrep[B3[1];S3M1[[1]]]0@12601AMatrixForm[urep[1;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.299 0 Td[(3]]MatrixForm[urep[1;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.298 0 Td[(3]]MatrixForm[urep[1;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.299 0 Td[(3]]0@13401Amatrep[B3[1];S3M1[[2]]]matrep[B3[1];S3M1[[2]]]matrep[B3[1];S3M1[[2]]]0@)]TJ/F19 11.955 Tf 9.298 0 Td[(2)]TJ/F19 11.955 Tf 9.298 0 Td[(1)]TJ/F19 11.955 Tf 9.298 0 Td[(631AMatrixForm[urep[)]TJ/F19 11.955 Tf 9.298 0 Td[(2;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.298 0 Td[(3]]MatrixForm[urep[)]TJ/F19 11.955 Tf 9.298 0 Td[(2;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.299 0 Td[(3]]MatrixForm[urep[)]TJ/F19 11.955 Tf 9.299 0 Td[(2;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.298 0 Td[(3]]0@)]TJ/F19 11.955 Tf 9.298 0 Td[(26131Amatrep[B3[1];S3M1[[3]]]matrep[B3[1];S3M1[[3]]]matrep[B3[1];S3M1[[3]]]0@)]TJ/F19 11.955 Tf 9.298 0 Td[(2)]TJ/F19 11.955 Tf 9.298 0 Td[(1)]TJ/F19 11.955 Tf 9.298 0 Td[(631AS3M2=SRG3modIrredTmodc2;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;S3M2=SRG3modIrredTmodc2;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;S3M2=SRG3modIrredTmodc2;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;Chop[N[matrep[B3[1];S3M2[[1]]]]]Chop[N[matrep[B3[1];S3M2[[1]]]]]Chop[N[matrep[B3[1];S3M2[[1]]]]] 115

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AppendixB:Continued 0@0:5320891:12:2490:4679111AMatrixForm[urep[0.5320888862379565;MatrixForm[urep[0.5320888862379565;MatrixForm[urep[0.5320888862379565;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.299 0 Td[(3]]LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(3;)]TJ/F19 11.955 Tf 9.298 0 Td[(3]]LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.299 0 Td[(3]]0@0:5320893:467913:532090:4679111A3.46791111376204333.53208888623795673.46791111376204333.53208888623795673.46791111376204333.532088886237956712:249Chop[N[matrep[B3[1];S3M2[[2]]]]]Chop[N[matrep[B3[1];S3M2[[2]]]]]Chop[N[matrep[B3[1];S3M2[[2]]]]]0@)]TJ/F19 11.955 Tf 9.298 0 Td[(0:6527041:10:92131:65271AMatrixForm[urep[)]TJ/F19 11.955 Tf 9.298 0 Td[(0.6527036446661388;MatrixForm[urep[)]TJ/F19 11.955 Tf 9.298 0 Td[(0.6527036446661388;MatrixForm[urep[)]TJ/F19 11.955 Tf 9.299 0 Td[(0.6527036446661388;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.299 0 Td[(3]]LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.298 0 Td[(3]]LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3;)]TJ/F19 11.955 Tf 9.299 0 Td[(3]]0@)]TJ/F19 11.955 Tf 9.298 0 Td[(0:6527044:65272:34731:65271A4.6527036446661392.3472963553338614.6527036446661392.3472963553338614.6527036446661392.34729635533386110:9213S2M1=Simplify[SRG2mod[IrredTmod[c2;1]]];S2M1=Simplify[SRG2mod[IrredTmod[c2;1]]];S2M1=Simplify[SRG2mod[IrredTmod[c2;1]]];matrep[B2[1];S2M1[[1]]]matrep[B2[1];S2M1[[1]]]matrep[B2[1];S2M1[[1]]]0@)]TJ/F19 11.955 Tf 9.298 0 Td[(11641AMatrixForm[urep[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(2;)]TJ/F19 11.955 Tf 9.298 0 Td[(2]]MatrixForm[urep[)]TJ/F19 11.955 Tf 9.298 0 Td[(1;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2;)]TJ/F19 11.955 Tf 9.299 0 Td[(2]]MatrixForm[urep[)]TJ/F19 11.955 Tf 9.299 0 Td[(1;LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2;)]TJ/F19 11.955 Tf 9.299 0 Td[(2]]0@)]TJ/F19 11.955 Tf 9.298 0 Td[(16141Amatrep[B2[1];S2M1[[3]]]matrep[B2[1];S2M1[[3]]]matrep[B2[1];S2M1[[3]]])]TJ/F19 11.955 Tf 9.298 0 Td[(2S2M2=SRG2modIrredTmodc2;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;S2M2=SRG2modIrredTmodc2;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;S2M2=SRG2modIrredTmodc2;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;Chop[N[matrep[B2[1];S2M2[[1]]]]]Chop[N[matrep[B2[1];S2M2[[1]]]]]Chop[N[matrep[B2[1];S2M2[[1]]]]] 116

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AppendixB:Continued 0@)]TJ/F19 11.955 Tf 9.298 0 Td[(1:1:6:4:1AChop[N[matrep[B2[1];S2M2[[2]]]]]Chop[N[matrep[B2[1];S2M2[[2]]]]]Chop[N[matrep[B2[1];S2M2[[2]]]]]0@)]TJ/F19 11.955 Tf 9.298 0 Td[(1:1:6:4:1AChop[N[matrep[B2[1];S2M2[[3]]]]]Chop[N[matrep[B2[1];S2M2[[3]]]]]Chop[N[matrep[B2[1];S2M2[[3]]]]])]TJ/F19 11.955 Tf 9.298 0 Td[(2:Chop[N[matrep[B2[1];S2M2[[4]]]]]Chop[N[matrep[B2[1];S2M2[[4]]]]]Chop[N[matrep[B2[1];S2M2[[4]]]]])]TJ/F19 11.955 Tf 9.298 0 Td[(2:S1M2=SRG1modIrredTmodc2;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;S1M2=SRG1modIrredTmodc2;)]TJ/F19 11.955 Tf 9.299 0 Td[(12=3;S1M2=SRG1modIrredTmodc2;)]TJ/F19 11.955 Tf 9.298 0 Td[(12=3;Simplify[matrep[B1[1];S1M2[[1]]]]Simplify[matrep[B1[1];S1M2[[1]]]]Simplify[matrep[B1[1];S1M2[[1]]]])]TJ/F19 11.955 Tf 9.298 0 Td[(1Simplify[matrep[B1[1];S1M2[[2]]]]Simplify[matrep[B1[1];S1M2[[2]]]]Simplify[matrep[B1[1];S1M2[[2]]]]6Simplify[matrep[B1[1];S1M2[[3]]]]Simplify[matrep[B1[1];S1M2[[3]]]]Simplify[matrep[B1[1];S1M2[[3]]]])]TJ/F19 11.955 Tf 9.298 0 Td[(1Simplify[matrep[B1[1];S1M2[[4]]]]Simplify[matrep[B1[1];S1M2[[4]]]]Simplify[matrep[B1[1];S1M2[[4]]]])]TJ/F19 11.955 Tf 9.298 0 Td[(1Simplify[matrep[B1[1];S1M2[[5]]]]Simplify[matrep[B1[1];S1M2[[5]]]]Simplify[matrep[B1[1];S1M2[[5]]]])]TJ/F19 11.955 Tf 9.298 0 Td[(1Simplify[matrep[B1[1];S1M2[[6]]]]Simplify[matrep[B1[1];S1M2[[6]]]]Simplify[matrep[B1[1];S1M2[[6]]]])]TJ/F19 11.955 Tf 9.298 0 Td[(1Verifycyclemoduleactionsa=AllCycleMod;*toolongtoshowoutput*a=AllCycleMod;*toolongtoshowoutput*a=AllCycleMod;*toolongtoshowoutput*prmd=PrimaryModule;prmd=PrimaryModule;prmd=PrimaryModule;md=2;*specifywhichcyclemodule*md=2;*specifywhichcyclemodule*md=2;*specifywhichcyclemodule*ZeroVectorsQ[v ]:=Union[Flatten[v]]===f0gZeroVectorsQ[v ]:=Union[Flatten[v]]===f0gZeroVectorsQ[v ]:=Union[Flatten[v]]===f0g 117

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AppendixB:Continued actonu 1,iMap[ZeroVectorsQ;Map[ZeroVectorsQ;Map[ZeroVectorsQ;fUnion[a[[md;1]]:Es[1]:A[1]:Es[1]+A[0]])]TJ -215.026 0.332 Td[(fUnion[a[[md;1]]:Es[1]:A[1]:Es[1]+A[0]])]TJ -215.027 -0.332 Td[(fUnion[a[[md;1]]:Es[1]:A[1]:Es[1]+A[0]])]TJ -215.823 -20.921 Td[(fprmd[[2]]g;fprmd[[2]]g;fprmd[[2]]g;a[[md;1]]:Es[1]:A[2]:Es[3])]TJ/F19 11.955 Tf 11.955 0 Td[(RotateRight[a[[md;3]]];a[[md;1]]:Es[1]:A[2]:Es[3])]TJ/F19 11.955 Tf 11.956 0 Td[(RotateRight[a[[md;3]]];a[[md;1]]:Es[1]:A[2]:Es[3])]TJ/F19 11.955 Tf 11.955 0 Td[(RotateRight[a[[md;3]]];a[[md;1]]:Es[1]:A[2]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;5]];a[[md;1]]:Es[1]:A[2]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;5]];a[[md;1]]:Es[1]:A[2]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;5]];a[[md;1]]:Es[1]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;6]];a[[md;1]]:Es[1]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;6]];a[[md;1]]:Es[1]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;6]];a[[md;1]]:Es[1]:A[3]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]];a[[md;1]]:Es[1]:A[3]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]];a[[md;1]]:Es[1]:A[3]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]];Union[a[[md;1]]:Es[1]:A[1]:Es[0]])-222(fprmd[[1]]gg]Union[a[[md;1]]:Es[1]:A[1]:Es[0]])-222(fprmd[[1]]gg]Union[a[[md;1]]:Es[1]:A[1]:Es[0]])-222(fprmd[[1]]gg]fTrue;True;True;True;True;Truegactonu2,iMap[ZeroVectorsQ;Map[ZeroVectorsQ;Map[ZeroVectorsQ;fUnion[a[[md;2]]:Es[2]:A[2]:Es[2]+A[0]])]TJ -215.026 0.332 Td[(fUnion[a[[md;2]]:Es[2]:A[2]:Es[2]+A[0]])]TJ -215.026 -0.332 Td[(fUnion[a[[md;2]]:Es[2]:A[2]:Es[2]+A[0]])]TJ -215.824 -20.922 Td[(fPrimaryModule[[3]]g;fPrimaryModule[[3]]g;fPrimaryModule[[3]]g;a[[md;2]]:Es[2]:A[3]:Es[1])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;1]];a[[md;2]]:Es[2]:A[3]:Es[1])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;1]];a[[md;2]]:Es[2]:A[3]:Es[1])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;1]];a[[md;2]]:Es[2]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;6]];a[[md;2]]:Es[2]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;6]];a[[md;2]]:Es[2]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;6]];a[[md;2]]:Es[2]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;4]];a[[md;2]]:Es[2]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;4]];a[[md;2]]:Es[2]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;4]];a[[md;2]]:Es[2]:A[1]:Es[3])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;3]]g]a[[md;2]]:Es[2]:A[1]:Es[3])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;3]]g]a[[md;2]]:Es[2]:A[1]:Es[3])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;3]]g]fTrue;True;True;True;Truegactonu3,iMap[ZeroVectorsQ;Map[ZeroVectorsQ;Map[ZeroVectorsQ;fUnion[a[[md;3]]:Es[3]:A[3]:Es[3]+A[0]])]TJ -215.027 0.332 Td[(fUnion[a[[md;3]]:Es[3]:A[3]:Es[3]+A[0]])]TJ -215.026 -0.332 Td[(fUnion[a[[md;3]]:Es[3]:A[3]:Es[3]+A[0]])]TJ -215.823 -20.922 Td[(fPrimaryModule[[4]]g;fPrimaryModule[[4]]g;fPrimaryModule[[4]]g;a[[md;3]]:Es[3]:A[1]:Es[2])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;2]];a[[md;3]]:Es[3]:A[1]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]];a[[md;3]]:Es[3]:A[1]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]];a[[md;3]]:Es[3]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;4]];a[[md;3]]:Es[3]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;4]];a[[md;3]]:Es[3]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;4]];a[[md;3]]:Es[3]:A[2]:Es[4])]TJ/F27 11.955 Tf -127.005 0.332 Td[(a[[md;3]]:Es[3]:A[2]:Es[4])]TJ/F27 11.955 Tf -127.004 -0.332 Td[(a[[md;3]]:Es[3]:A[2]:Es[4])]TJ/F19 11.955 Tf -127.801 -20.922 Td[(RotateLeft[a[[md;5]]];RotateLeft[a[[md;5]]];RotateLeft[a[[md;5]]];a[[md;3]]:Es[3]:A[2]:Es[1])]TJ/F19 11.955 Tf 11.956 0 Td[(RotateLeft[a[[md;1]]]g]a[[md;3]]:Es[3]:A[2]:Es[1])]TJ/F19 11.955 Tf 11.955 0 Td[(RotateLeft[a[[md;1]]]g]a[[md;3]]:Es[3]:A[2]:Es[1])]TJ/F19 11.955 Tf 11.955 0 Td[(RotateLeft[a[[md;1]]]g] 118

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AppendixB:Continued fTrue;True;True;True;Truegactonv1,iMap[ZeroVectorsQ;Map[ZeroVectorsQ;Map[ZeroVectorsQ;fa[[md;4]]:Es[4]:A[1]:Es[4]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3)]TJ -255.709 0.332 Td[(fa[[md;4]]:Es[4]:A[1]:Es[4]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3)]TJ -255.71 -0.332 Td[(fa[[md;4]]:Es[4]:A[1]:Es[4]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3)]TJ/F27 11.955 Tf -256.506 -20.922 Td[(a[[md;4]];a[[md;4]];a[[md;4]];Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.289 0.333 Td[(Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.29 -0.333 Td[(Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F27 11.955 Tf -216.086 -20.921 Td[(a[[md;6]]]+fPrimaryModule[[5]]g;a[[md;6]]]+fPrimaryModule[[5]]g;a[[md;6]]]+fPrimaryModule[[5]]g;Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[1])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;1]]]+Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[1])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;1]]]+Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[3]:Es[1])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;1]]]+fPrimaryModule[[2]]g;fPrimaryModule[[2]]g;fPrimaryModule[[2]]g;Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]]]+Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]]]+Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[3]:Es[2])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;2]]]+fPrimaryModule[[3]]g;fPrimaryModule[[3]]g;fPrimaryModule[[3]]g;a[[md;4]]:Es[4]:A[1]:Es[2]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf -258.836 0.332 Td[(a[[md;4]]:Es[4]:A[1]:Es[2]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(2)]TJ/F27 11.955 Tf -258.837 -0.332 Td[(a[[md;4]]:Es[4]:A[1]:Es[2]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf -259.634 -20.921 Td[(a[[md;2]];a[[md;2]];a[[md;2]];a[[md;4]]:Es[4]:A[1]:Es[3]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf -258.836 0.332 Td[(a[[md;4]]:Es[4]:A[1]:Es[3]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(2)]TJ/F27 11.955 Tf -258.837 -0.332 Td[(a[[md;4]]:Es[4]:A[1]:Es[3]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf -259.633 -20.921 Td[(a[[md;3]];a[[md;3]];a[[md;3]];Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[3])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;3]]]+Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[3])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;3]]]+Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[2]:Es[3])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;3]]]+fPrimaryModule[[4]]g;fPrimaryModule[[4]]g;fPrimaryModule[[4]]g;Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[1])]TJ/F19 11.955 Tf -179.545 0.332 Td[(Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[1])]TJ/F19 11.955 Tf -179.546 -0.332 Td[(Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[2]:Es[1])]TJ/F19 11.955 Tf -180.343 -20.922 Td[(RotateLeft[a[[md;1]]]]+fPrimaryModule[[2]]g;RotateLeft[a[[md;1]]]]+fPrimaryModule[[2]]g;RotateLeft[a[[md;1]]]]+fPrimaryModule[[2]]g;Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.289 0.332 Td[(Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.29 -0.332 Td[(Union[a[[md;4]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[2]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -216.086 -20.922 Td[(RotateLeft[a[[md;5]]]]+fPrimaryModule[[5]]gg]RotateLeft[a[[md;5]]]]+fPrimaryModule[[5]]gg]RotateLeft[a[[md;5]]]]+fPrimaryModule[[5]]gg]fTrue;True;True;True;True;True;True;True;Truegactonv2,iMap[ZeroVectorsQ;Map[ZeroVectorsQ;Map[ZeroVectorsQ;fa[[md;5]]:Es[4]:A[2]:Es[4]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3)]TJ -255.709 0.332 Td[(fa[[md;5]]:Es[4]:A[2]:Es[4]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3)]TJ -255.71 -0.332 Td[(fa[[md;5]]:Es[4]:A[2]:Es[4]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3)]TJ/F27 11.955 Tf -256.506 -20.921 Td[(a[[md;5]];a[[md;5]];a[[md;5]];Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.289 0.332 Td[(Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.29 -0.332 Td[(Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -216.086 -20.922 Td[(RotateRight[a[[md;4]]]]+fPrimaryModule[[5]]g;RotateRight[a[[md;4]]]]+fPrimaryModule[[5]]g;RotateRight[a[[md;4]]]]+fPrimaryModule[[5]]g; 119

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AppendixB:Continued Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[2])]TJ/F19 11.955 Tf -179.545 0.332 Td[(Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[2])]TJ/F19 11.955 Tf -179.546 -0.332 Td[(Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[1]:Es[2])]TJ/F19 11.955 Tf -180.343 -20.921 Td[(RotateRight[a[[md;2]]]]+fPrimaryModule[[3]]g;RotateRight[a[[md;2]]]]+fPrimaryModule[[3]]g;RotateRight[a[[md;2]]]]+fPrimaryModule[[3]]g;Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[3])]TJ/F19 11.955 Tf -179.545 0.332 Td[(Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[1]:Es[3])]TJ/F19 11.955 Tf -179.546 -0.332 Td[(Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[1]:Es[3])]TJ/F19 11.955 Tf -180.343 -20.921 Td[(RotateRight[a[[md;3]]]]+fPrimaryModule[[4]]g;RotateRight[a[[md;3]]]]+fPrimaryModule[[4]]g;RotateRight[a[[md;3]]]]+fPrimaryModule[[4]]g;a[[md;5]]:Es[4]:A[2]:Es[3]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf -258.836 0.332 Td[(a[[md;5]]:Es[4]:A[2]:Es[3]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(2)]TJ/F27 11.955 Tf -258.837 -0.332 Td[(a[[md;5]]:Es[4]:A[2]:Es[3]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F19 11.955 Tf -259.633 -20.922 Td[(RotateRight[a[[md;3]]];RotateRight[a[[md;3]]];RotateRight[a[[md;3]]];a[[md;5]]:Es[4]:A[2]:Es[2]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf -258.836 0.332 Td[(a[[md;5]]:Es[4]:A[2]:Es[2]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(2)]TJ/F27 11.955 Tf -258.837 -0.332 Td[(a[[md;5]]:Es[4]:A[2]:Es[2]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf -259.633 -20.922 Td[(a[[md;1]];a[[md;1]];a[[md;1]];Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[1])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;1]]]+Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[1])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;1]]]+Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[3]:Es[1])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;1]]]+fPrimaryModule[[2]]g;fPrimaryModule[[2]]g;fPrimaryModule[[2]]g;Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]]]+Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]]]+Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[3]:Es[2])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;2]]]+fPrimaryModule[[3]]g;fPrimaryModule[[3]]g;fPrimaryModule[[3]]g;Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.289 0.332 Td[(Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.29 -0.332 Td[(Union[a[[md;5]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[3]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F27 11.955 Tf -216.086 -20.922 Td[(a[[md;6]]]+fPrimaryModule[[5]]gg]a[[md;6]]]+fPrimaryModule[[5]]gg]a[[md;6]]]+fPrimaryModule[[5]]gg]fTrue;True;True;True;True;False;True;True;Truegactonv3,iMap[ZeroVectorsQ;Map[ZeroVectorsQ;Map[ZeroVectorsQ;fa[[md;6]]:Es[4]:A[3]:Es[4]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3)]TJ -255.709 0.332 Td[(fa[[md;6]]:Es[4]:A[3]:Es[4]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(3)]TJ -255.71 -0.332 Td[(fa[[md;6]]:Es[4]:A[3]:Es[4]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(3)]TJ/F27 11.955 Tf -256.507 -20.922 Td[(a[[md;6]];a[[md;6]];a[[md;6]];Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.289 0.332 Td[(Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.29 -0.332 Td[(Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[2]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F27 11.955 Tf -216.086 -20.922 Td[(a[[md;5]]]+fPrimaryModule[[5]]g;a[[md;5]]]+fPrimaryModule[[5]]g;a[[md;5]]]+fPrimaryModule[[5]]g;Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[3])]TJ/F19 11.955 Tf -179.546 0.332 Td[(Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[3])]TJ/F19 11.955 Tf -179.546 -0.332 Td[(Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[2]:Es[3])]TJ/F19 11.955 Tf -180.343 -20.922 Td[(RotateRight[a[[md;3]]]]+fPrimaryModule[[4]]g;RotateRight[a[[md;3]]]]+fPrimaryModule[[4]]g;RotateRight[a[[md;3]]]]+fPrimaryModule[[4]]g;Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[1])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;1]]]+Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[2]:Es[1])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;1]]]+Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[2]:Es[1])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;1]]]+fPrimaryModule[[2]]g;fPrimaryModule[[2]]g;fPrimaryModule[[2]]g;a[[md;6]]:Es[4]:A[3]:Es[1]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf -258.836 0.332 Td[(a[[md;6]]:Es[4]:A[3]:Es[1]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(2)]TJ/F27 11.955 Tf -258.837 -0.332 Td[(a[[md;6]]:Es[4]:A[3]:Es[1]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf -259.634 -20.922 Td[(a[[md;1]];a[[md;1]];a[[md;1]];a[[md;6]]:Es[4]:A[3]:Es[2]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F27 11.955 Tf -258.836 0.333 Td[(a[[md;6]]:Es[4]:A[3]:Es[2]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.956 0 Td[(2)]TJ/F27 11.955 Tf -258.837 -0.333 Td[(a[[md;6]]:Es[4]:A[3]:Es[2]=LSToStudyOrder)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJET1 0 0 1 99.895 64.458 cm0 g 0 G1 0 0 1 -99.895 -64.458 cmBT/F19 11.955 Tf 316.77 64.458 Td[(120

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AppendixB:Continued a[[md;2]];a[[md;2]];a[[md;2]];Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]]]+Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[2])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;2]]]+Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[1]:Es[2])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;2]]]+fPrimaryModule[[3]]g;fPrimaryModule[[3]]g;fPrimaryModule[[3]]g;Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[3])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;3]]]+Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[3])]TJ/F27 11.955 Tf 11.955 0 Td[(a[[md;3]]]+Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[1]:Es[3])]TJ/F27 11.955 Tf 11.956 0 Td[(a[[md;3]]]+fPrimaryModule[[4]]g;fPrimaryModule[[4]]g;fPrimaryModule[[4]]g;Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.289 0.333 Td[(Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.299 0 Td[(Es[4]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F19 11.955 Tf -215.29 -0.333 Td[(Union[a[[md;6]]:)]TJ/F19 11.955 Tf 9.298 0 Td[(Es[4]:A[1]:Es[4])]TJ/F27 11.955 Tf 11.955 0 Td[(A[0])]TJ/F27 11.955 Tf -216.086 -20.921 Td[(a[[md;4]]]+fPrimaryModule[[5]]gg]a[[md;4]]]+fPrimaryModule[[5]]gg]a[[md;4]]]+fPrimaryModule[[5]]gg]fTrue;True;True;True;True;True;True;True;Trueg 121

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AbouttheAuthorIbtisamDaqqareceivedaBachelorsDegreeinMathematicsfromAppliedScienceUniver-sityin1995andaMaster'sDegreeinMathematicsfromUniversityofJordanin1998.In1999shestartedteachingasaninstructorofMathematicsattheHashemiteUniversityinJordan.SheenteredthePh.D.programasateachingassistantattheUniversityofSouthFloridain2002.WhileinthePh.D.programattheUniversityofSouthFlorida,Mrs.Daqqataughtseveralundergraduateclassesandsubmittedtwopapersforpublication.ShealsomadeseveralpresentationsatAMSsectionalandnationalmeetings.


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Daqqa, Ibtisam.
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Subconstituent algebras of Latin squares
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by Ibtisam Daqqa.
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[Tampa, Fla] :
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2008.
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Dissertation (Ph.D.)--University of South Florida, 2008.
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ABSTRACT: It is well-known that one may construct a 4-class association scheme on the positions of a Latin square, where the relations are the identity, being in the same row, being in the same column, having the same entry, and everything else. We describe the subconstituent (Terwilliger) algebras of such an association scheme. One also may construct several strongly regular graphs on the positions of a Latin square, where adjacency corresponds to any subset of the nonidentity relations described above. We describe the local spectrum and subconstituent algebras of such strongly regular graphs. Finally, we study various notions of isomorphism for subconstituent algebras using Latin squares as examples.
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Advisor: Brian Curtin, Ph.D.
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Terwilliger Algebra
Strongly regular graph
Association scheme
Bose -Mesner Algebra
Fusions
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