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Some combinatorial structures constructed from modular Leonard triples

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Some combinatorial structures constructed from modular Leonard triples
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Sobkowiak, Jessica
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Leonard pairs
Antiautomorphisms
Valid sequence
Simplicial complex
Hamming association scheme
Dissertations, Academic -- Mathematics and Statistics -- Masters -- USF   ( lcsh )
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ABSTRACT: Let V denote a vector space of finite positive dimension. An ordered triple of linear operators on V is said to be a Leonard triple whenever for each choice of element of the triple there exists a basis of V with respect to which the matrix representing the chosen element is diagonal and the matrices representing the other two elements are irreducible tridiagonal. A Leonard triple is said to be modular whenever for each choice of element there exists an antiautomorphism of End(V) which fixes the chosen element and swaps the other two elements. We study combinatorial structures associated with Leonard triples and modular Leonard triples. In the first part we construct a simplicial complex of Leonard triples. The simplicial complex of a Leonard triple is the smallest set of linear operators which contains the given Leonard triple with the property that if two elements of the set are part of a Leonard triple, then the third element of the triple is also in the set. In the second part we construct a Hamming association scheme from modular Leonard triples using a method used previously in the context of Grassmanian codes.
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Thesis (M.A.)--University of South Florida, 2009.
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by Jessica Sobkowiak.
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SomeCombinatorialStructuresConstructedfromModularLeonardTriples by JessicaSobkowiak Athesissubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofArts DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:BrianCurtin,Ph.D. MohamedElhamdadi,Ph.D. Xiang-dongHou,Ph.D. StephenSuen,Ph.D. DateofApproval: May6,2009 Keywords:Leonardpairs,Antiautomorphisms,Validsequence,Simplicialcomplex, Hammingassociationscheme c Copyright2009,JessicaSobkowiak

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Dedication Tomyincredibleparents,andmyamazingbrotherandsister,JeremyandSara.

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Acknowledgements IwouldliketosincerelythankDr.BrianCurtinforhisguidance,understanding, andpatiencethroughoutthisprocess.IwouldalsoliketothanktheDepartment ofMathematicsatUniversityofSouthFlorida,mythesiscommittee,Dr.Saito,and theadministrativesta,especiallySarinaMaldonadoandMaryAnnWengyn,for theirkindnessandassistance.ManythankstomyfriendsandclassmatesatUSFfor oeringtheirhumorandsupportthesepasttwoyears.Ithankmyparents,forhaving faithinmeevenwhenmyowncondencewavered.Ialsothankmybrotherandmy sisterforalwayslendingalisteningearandanencouragingword.AboveallIthank Godwhogavemethestrengthandinsightthatcomfortedmeinthewholeprocess ofthisthesis.

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TableofContents ListofTablesii ListofFiguresiii Abstractiv 1Introduction1 2PrelimariesforLeonardpairs4 2.1Endomorphismsandmatrices....................4 2.2Leonardpairsandtriples......................6 2.3Antiautomorphisms..........................7 2.4ModularLeonardtriples.......................8 2.5Thesenseofthetermmodular...................9 2.6CanonicalmodularLeonardtriples.................10 2.7Theautomorphismsandantiautomorphisms............15 2.8 Mathematica code:Generalimplementationsetup.........19 3 -orbitsofLeonardtriples22 3.1Validsequencesofantiautomorphisms...............22 3.2 -orbitsfromvalidsequences....................25 3.3 Mathematica code:ApplyingValidSequences...........28 3.4Thecomplexofan -orbit......................29 3.5Symmetryofan -orbit'scomplex.................31 3.6Vertex -orbitgrowth........................36 3.7Channel -orbitgrowth.......................39 i

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3.8Problems...............................43 4 -orbitsofmodularLeonardtriples44 4.1TypesII,IV,V,andVI.......................44 4.2TypesOandIII...........................51 4.3TypeI.................................61 4.4Problems...............................67 5AssociationschemesfrommodularLeonardtriples68 5.1Preliminariesforassociationschemes................68 5.2ThePaulimatrices..........................70 5.3APartitionofKroneckerproducts.................72 5.4HammingSchemesfrommodularLeonardtriples.........74 5.5Counterexamplesfornon-modulartriples..............76 5.6Problems...............................77 References78 AbouttheAuthorEndPage ii

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ListofTables Table3.1Beginningstagesof -orbitgrowth................27 iii

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ListofFigures Figure3.1Counterexampleofdistance-transitivity/regularity.......33 Figure3.2AntiautomorphismactiononLeonardtriples..........35 Figure3.3Vertex -orbitgrowth.......................36 Figure3.4Completetripartitegraph K 3 ; 2 ; 3 .................38 Figure3.5Channel -orbitgrowth......................39 Figure3.66possiblechannelsfromaLeonardtriple............40 Figure4.1TypeO/IIIcomplex 2 =1...................52 Figure4.2TypeO/IIIcomplex 3 =1...................54 Figure4.3TypeO/IIIcomplex 4 =1...................55 Figure4.4TypeO/IIIcomplex 5 =1...................57 Figure4.5Conjecturedinnitecomplex 7 =1..............59 Figure4.6Non-regularsolidswithtriangularfaces.............60 Figure4.7TypeIcomplex 6 = q 6 =1...................63 Figure4.8TypeIcomplex 7 = q 7 =1...................65 Figure5.1The3-dimensionalbinaryHammingcube H ; 2........70 iv

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SomeCombinatorialStructuresConstructed fromModularLeonardTriples JessicaSobkowiak Abstract Let V denoteavectorspaceofnitepositivedimension.Anorderedtripleoflinear operatorson V issaidtobeaLeonardtriplewheneverforeachchoiceofelementof thetriplethereexistsabasisof V withrespecttowhichthematrixrepresenting thechosenelementisdiagonalandthematricesrepresentingtheothertwoelements areirreducibletridiagonal.ALeonardtripleissaidtobemodularwheneverfor eachchoiceofelementthereexistsanantiautomorphismofEnd V whichxesthe chosenelementandswapstheothertwoelements.Westudycombinatorialstructures associatedwithLeonardtriplesandmodularLeonardtriples.Intherstpartwe constructasimplicialcomplexofLeonardtriples.ThesimplicialcomplexofaLeonard tripleisthesmallestsetoflinearoperatorswhichcontainsthegivenLeonardtriple withthepropertythatiftwoelementsofthesetarepartofaLeonardtriple,then thethirdelementofthetripleisalsointheset.Inthesecondpartweconstruct aHammingassociationschemefrommodularLeonardtriplesusingamethodused previouslyinthecontextofGrassmaniancodes. v

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1Introduction InthisthesiswestudysomecombinatorialstructuresconstructedfromLeonard triples.Weshallrecallbackgroundmaterial,includingprecisedenitions,inChapter 2.ForthesakeofdiscussionwenotethatLeonardpairsandtriplesarerespectively orderedpairsandtriplesoflinearoperatorsandthatanytwoelementsofaLeonard tripleformaLeonardpair. MuchofourworkfocusesonaspecialfamilyofLeonardtriplescalledmodular Leonardtriples.In[13]themodularLeonardtripleswerecompletelyclassiedinto sevenfamilies.In[12],[13],connectionstothemodulargroupandthusthebraid groupandtospinmodelsforlinkinvariantsweredeveloped.Arelatedfamilyof Leonardpairsalsoarisesinconnectionwithspinmodelsondistance-regulargraphs [11],[12]. IntherstpartofthisthesisChapters3and4weconstructasimplicialcomplex fromaLeonardtriple.WeidentifyLeonardtripleswithtriangularfaces,Leonardpairs withedges,andtheelementsofLeonardtripleswithpoints.Thesimplicialcomplex isthesmallestsetoflinearoperatorswhichcontainsthegivenLeonardtriplewiththe propertythatiftwoelementsofthesetarepartofaLeonardtriple,thenthethird elementofthetripleisalsointheset.InChapter3wedevelopsometheoreticalresults andtoolstostudythiscomplex.Inparticular,thisprovidesameansofconstructing thesimplicialcomplexofamodularLeonardtriple. InChapter4,westudythesimplicialcomplexofthemodularLeonardtriples.We showthatfor4ofthe7familiesofmodularLeonardtriples,theassociatedcomplexis justasingletriangle.Fortheremaining3families,theassociatedsimplicialcomplex appearstobeinniteunlesstheparameterstakeonspecialvalues.Ourfocusisonthe casewhenthesimplicialcomplexisnite.Wendexamplesinwhichthecomplex 1

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isatetrahedron,anoctahedron,anicosahedron,andatriangulartorus.Wealso ndexampleswhichhaveacertainlocalnitenessproperty,butwhichnonetheless appeartobeinnite.Theworkinthischapteriscomputational,soweincludesome Mathematica codetoillustratethecomputations. InthesecondpartofthisthesisChapter5weshowthatonemayconstruct aHammingassociationschemefromModularLeonardtriplesusingamethodused previouslyinthecontextofGrassmaniancodesbyRoy[39],[40].ThePaulimatrices areacommonexamplefromthesetwotheories. Werecallalittlehistory.LeonardpairswereintroducedbyP.Terwilliger[41]to completeE.BannaiandT.Ito's[4]algebraicabstractionoftheworkofD.Leonard [24].Leonardgaveacharacterizationoftheorthogonalpolynomialswithnitesupportwhosedualsarealsoorthogonalpolynomialswithnitesupport[24].Leonard's workaroseinthecontextofassociationschemes{anaturalcontextatthetime.Indeed,inhisthesis[15],Delsartedescribedacertaindualityinmetricandco-metric P -and Q -polynomialassociationschemesusingknownorthogonalpolynomialsand varioushypergeometricfunctions[16]whichbehavedsimilarly.Itwasthisworkwhich inspiredR.AskeyandJ.Wilsontofurtherstudytheconnectionbetweenhypergeometricfunctionsandorthogonalpolynomials[2],rstleadinghimtodiscoverthe q -Racahpolynomialsamongtheorthogonalpolynomialswithnitesupport,andlater totheclassicationbyR.AskeyandJ.Wilsonofallorthogonalpolynomialswith orthogonalduals[1].TheresultsofLeonardandofAskeyandWilson[1],[2]are analogousandindependent,theformertreatingthecasewithdiscretesupport,the lattertreatingthecasewithcontinuoussupport.Theorthogonalpolynomialsrelated toLeonardpairsformtheterminatingbranchoftheAskeyscheme[52]. Leonardpairsarealsorelatedtoassociationschemes.Amongassociationschemes, someofthenicestarethosearisingfromdistance-regulargraphsthemetric,or P-polynomialassociationsschemes[4],[5].Fromanassociationschemeonemay constructasubconstituent,orTerwilliger,algebra[9].Theirreduciblerepresentations ofthesubconstituentalgebraofadistance-regulargrapharetridiagonalpairs[41], [42],[43].InmanyinstancestheyareactuallyLeonardpairs[19].Wenotethat iftheassociationschemeofadistance-regulargraphsupportsaspinmodel,then 2

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everyirreduciblerepresentationoftheassociatedsubconstituentalgebraisnotonly aLeonardpair[10],butaspinLeonardpair[9].SpinLeonardpairsareknownto extendtoamodularLeonardtriple[12].Wenotethattheknownniteexamplesof simplicialcomplexeshaveanassociationschemestructure. TheLeonardpairsarecompletelyclassiedinto13familiesaccordingtowhich orthogonalpolynomialsarisefromthem[52].Theirconnectiontoorthogonalpolynomialshasbeenfurtherdevelopedin[6],[7],[18],[19],[26],[36],[41],[42],[43],[44], [45],[46],[47],[48],[49],[51],[52],[53],[54],[55],and[56]. Theseorthogonalpolynomialsarisefrequentlyinconnectionwithnite-dimensional representationofmanynicegroups,Liealgebras,andquantumgroups.Thusitisno surprisetondLeonardpairsassociatedwithvariousalgebras[6],[7],[9],[17],[36], [41],[42],[43].Leonardpairshavealsobeenstudiedaslinearalgebraicobjectsin theirownrightaswell[12],[13],[17],[26],[27],[28],[29],[30],[31],[32],[33],[34][35], [44],[45],[46],[47],[48],[49],[52],[53],[54],and[55]. 3

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2PrelimariesforLeonardpairs Inthischapterwerecallsomematerialconcerningendomorphisms,antiautomorphisms,Leonardpairs,Leonardtriples,andrelatedobjects.Wewillusethismaterial inourworkinthefollowingchapter. 2.1Endomorphismsandmatrices Henceforth,let K denoteaeld,andlet V denoteavectorspaceover K ofnite positivedimension. Denition2.1.1 Bya linearoperatoron V wemeana K -linearmapfrom V to V LetEnd V denotethe K -algebraconsistingofalllinearoperatorson V Itiswell-knownthatlinearoperatorsonnite-dimensionalvectorspacesmaybe identiedwithmatrices.LetMat n K bethe K -algebraofall n n matriceswith entriesin K Theorem2.1.2 [22]Let 2 End V .Supposethat 1 ::: n isanorderedbasis B of V .Thentherearescalars a ij 2 K i;j n suchthatforalli, i = a i 1 1 + ::: + a in n : The n n matrix a ij 2 Mat n K iscalledthe matrixof relativetothebasis B andisdenoted [ ] B .Themapfrom End V into Mat n K denedby 7! [ ] B isa K -algebraisomorphism. Denition2.1.3 Bya K -algebra automorphism ofEnd V ,wemeana K -algebra isomorphismfromEnd V toEnd V 4

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Denition2.1.4 Byan antiautomorphism ofEnd V ,wemeana K -linearbijection :End V End V suchthat XY = Y X forall X;Y 2 End V Matricesprovideconcretedescriptionsofautomorphismsandantiautomorphisms. Lemma2.1.5 [38]Amap : Mat n K Mat n K isa K -algebraautomorphismif andonlyifthereexistsaninvertible S 2 Mat n K suchthat X = SXS )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 forall X 2 Mat n K .Inthiscasewesaythat S represents Lemma2.1.6 Let denotea K -algebraautomorphismof Mat n K .Assumethat S 2 Mat n K represents .Thenfor S 0 2 Mat n K thefollowingareequivalent: i S 0 represents ii Thereexistsanonzeroscalar a 2 K suchthat S 0 = aS Lemma2.1.7 [38]Amap : Mat n K Mat n K isa K -algebraantiautomorphism ifandonlyifthereexistsaninvertible S 2 Mat n K suchthat X = SX t S )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 forall X 2 Mat n K ;where X t isthetransposeof X .Inthiscasewesaythat S represents Lemma2.1.8 Let denotea K -algebraantiautomorphismof Mat n K .Assumethat S 2 Mat n K represents .Thenfor S 0 2 Mat n K thefollowingareequivalent: i S 0 represents ii Thereexistsanonzeroscalar a 2 K suchthat S 0 = aS 5

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2.2Leonardpairsandtriples InthissectionwerecallthenotionsofLeonardpairsandtriples. Denition2.2.1 Asquarematrixover K issaidtobe tridiagonal wheneverevery nonzeroentryappearsonthediagonal,thesuperdiagonal,orthesubdiagonal.A tridiagonalmatrixis irreducible whenevertheentriesonthesubdiagonalsandsuperdiagonalsareallnonzero. Denition2.2.2 Let A 1 A 2 denoteanorderedpairofelementstakenfromEnd V Wecallthispaira LeonardpaironV wheneverforeach B 2f A 1 ;A 2 g ,thereexistsa basisof V withrespecttowhichthematrixrepresenting B isdiagonalandthematrix representingtheothermemberofthepairisirreducibletridiagonal. Lemma2.2.3 [45]Let A 1 A 2 denoteaLeonardpair.Then A 1 and A 2 together generate End V Denition2.2.4 Let A 1 A 2 A 3 denoteanorderedtripleofelementstakenfrom End V .Wecallthistriplea LeonardtripleonV wheneverforeach B 2f A 1 ;A 2 ;A 3 g thereexistsabasisof V withrespecttowhichthematrixrepresenting B isdiagonal andthematricesrepresentingtheothertwomembersofthetripleareirreducible tridiagonal.Anyorderedpairofdistinctelementsof A 1 A 2 A 3 formaLeonardpair. Denition2.2.5 GivenaLeonardtripleon V ,werefertodim V -1asits diameter TheLeonardpairsarecompletelyclassiedinto13familiesaccordingtowhich orthogonalpolynomialsarisefromthem[52].Theirconnectiontoorthogonalpolynomialshasbeenfurtherdevelopedin[6],[7],[18],[19],[26],[36],[41],[42],[43], [44],[45],[46],[47],[48],[49],[51],[52],[53],[54],[55],and[56].Theseorthogonal polynomialsarisefrequentlyinconnectionwithnite-dimensionalrepresentationof manynicegroups,Liealgebras,andquantumgroups[6],[7],[9],[17],[36],[41],[42], [43].Leonardpairshavealsobeenstudiedaslinearalgebraicobjectsintheirown rightaswell[12],[13],[17],[26],[27],[28],[29],[30],[31],[32],[33],[34][35],[44],[45], [46],[47],[48],[49],[52],[53],[54],and[55]. 6

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2.3Antiautomorphisms WerecallaspecialantiautomorphismassociatedwithaLeonardpair. Lemma2.3.1 [48]Let A 1 A 2 denoteaLeonardpair.Thenthereexistsanantiautomorphismof End V whichxesboth A 1 and A 2 .Thisantiautomorphismisunique andaninvolution. Lemma2.3.2 Let A 1 A 2 A 3 beaLeonardtriple.Foreachpermutation ijk of 123 let jk i betheantiautomorphismfromLemma2.3.1xing A j and A k .Then 23 1 A 1 A 2 A 3 ; A 1 13 2 A 2 A 3 ; A 1 A 2 12 3 A 3 arealsoLeonardtriples. Proof. SinceanytwoelementsofaLeonardtripleformaLeonardpair,byLemma 2.3.1weknowthatsuchan jk i existssuchthat jk i A j = A j and jk i A k = A k .Since antiautomorphismsdonotchangetheunderlyingshapeofthematrix,irreducible tridiagonalordiagonal,itfollowsfromDenition2.2.4thateach jk i A i A j A k is alsoaLeonardtriple. 2 WeshallmeetseveralfamiliesofLeonardtriplesforwhichalloftheLeonardtriples ofLemma2.3.2coincide. Denition2.3.3 Let a 1 a 2 a 3 benonzeroscalarsandlet b 1 b 2 b 3 beanyscalars. Let I betheidentitytransformation.WiththenotationofLemma2.3.2,wereferto anytripleoftheform a 1 A 1 + b 1 I a 2 A 2 + b 2 I a 3 A 3 + b 3 I asan anetransformation oftheoriginaltriple A 1 A 2 A 3 ObservethatananetransformationofaLeonardtripleisalsoaLeonardtriple. ThisallowsustoconsiderfewerparameterswhendiscussingtypesofmodularLeonard triplesinlaterchapters.Observethatanyantiautomorphismxingtwoof A 1 A 2 A 3 alsoxesthecorrespondingtwoof a 1 A 1 + b 1 I a 2 A 2 + b 2 I a 3 A 3 + b 3 I Lemma2.3.4 [14]Uptoaneequivalence,everyLeonardpairisinatmosttwo Leonardtriples. Lemma2.3.5 [14]If A 1 A 2 A 3 and A 1 A 2 A 0 3 aredistinctLeonardtriples,then a 12 3 A 3 + bI = A 0 3 forsome a b 2 K 7

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2.4ModularLeonardtriples InthissectionwerecallthenotionofmodularLeonardtriples. Denition2.4.1 [13]Let A 1 A 2 A 3 denoteaLeonardtripleon V .Thenthis Leonardtripleissaidtobe modular wheneverforeach B 2f A 1 ;A 2 ;A 3 g thereexists anantiautomorphismofEnd V whichxes B andswapstheothertwomembersof thetriple. Lemma2.4.2 [13]Let A 1 A 2 A 3 beamodularLeonardtriple.Foreachpermutation ijk of 123 ,let jk i betheantiautomorphismfromDenition2.4.1whichxes A i and swaps A j and A k .Let ij = ik j jk i and 0 ij = jk i ij k .Then ij A j = A k = 0 ij A j ; .4.1 ij A i = A i = 0 ij A i ; .4.2 ij A k = j A j = 0 ij A k : .4.3 Inparticular, ij = 0 ij Proof. Bydenition, ij A j = ik j jk i A j = ik j A k = A k ij A i = ik j jk i A i = ik j A i = A i 0 ij A j = i k A j = i A j = A k 0 ij A i = jk i ij k A i = jk i A i = A i ,and ij A k = ik j jk i A k = ik j A j .Inthesameway, 0 ij A k = jk i ij k A k = ik j A j .Thus,theresultfollowsfromLemma2.2.3. 2 Corollary2.4.3 WithreferencetoLemma2.4.2, )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ij = ik Proof. ByLemma2.4.2, ij ik A k = ij A j = A k .Similarly, ij ik A i = ij A i = A i .Thus,theresultholds. 2 Notethat 23 1 13 2 12 3 inducean S 3 actionon A 1 A 2 A 3 bypermutation,where S 3 isthesymmetricgroupoforder3. 8

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2.5Thesenseofthetermmodular Inthissection,weexplainthattheterm modular inmodularLeonardtriplesarises fromaconnectionwiththemodulargroup.DenotebySL 2 Z thesetofall2 2 matrices M withentriesin Z suchthatdet M =1.ThenSL 2 Z isagroupunder matrixmultiplication,andiscalledthe2-dimensional speciallineargroup over Z [22].NotethatthecenterofSL 2 Z is f I 2 2 ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(I 2 2 g Denition2.5.1 [22]The modulargroup isthegroupPSL 2 Z =SL 2 Z / f I 2 2 g PSL 2 Z isalsocalledthetwo-dimensional projectivespeciallineargroup Theorem2.5.2 [22]PSL 2 Z haspresentation h a;b j a 2 =1 ;b 3 =1 i TheusualcorrespondencebetweenDenition2.5.1andTheorem2.5.2isgivenby a = 0 @ 0 )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 10 1 A and b = 0 @ 0 )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 11 1 A Lemma2.5.3 WiththenotationofLemma2.4.2andDenition2.5.1, PSL 2 Z acts on End V asagroupofautomorphismson End V whereaactsas ij jk ij andb actsas ij jk Proof. ByCorollary2.4.3, ij jk ij 2 = ik ij ik ij =1and ij jk 3 = ik 3 =1. 2 Denition2.5.4 GivenaLeonardtriple A 1 A 2 A 3 ,wedeneanotherLeonardtriple inthesamePSL 2 Z orbitbytheactionof a and b andwemaydeneanotherPSL 2 Z actionviathecorrespondencebetweenanyotherLeonardtripleinthePSL 2 Z orbit includingcertainpermutationsof A i A j A k bytheextendedorbit. 9

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2.6CanonicalmodularLeonardtriples ModularLeonardtripleshavebeenclassieduptoisomorphism[13].Thisclassicationgivesamatrixrepresentativeofeachisomorphismclass.Thisconcretedescription isusefulwhencomputingwithmodularLeonardtriples.Thefollowingnotationand conventionsareusedtoclassifymodularLeonardtriples[13].Therowsandcolumns ofmatricesinMat d +1 K shallbeindexedby0,1, ::: d Denition2.6.1 [13]Bya canonicalmodularLeonardtripleofdiameter d ,wemean anorderedtripleofmatrices A 1 A 2 A 3 fromMat d +1 K whichformamodular Leonardtripleon K d +1 andsuchthati A 2 isdiagonal;iieachof A 1 A 3 isirreducibletridiagonal;iiieachrowsumof A 1 equalstherstdiagonalentryof A 2 Lemma2.6.2 [13]Let A 1 A 2 A 3 denoteacanonicalmodularLeonardtripleof diameterd.Then A 1 = tridiag 0 B B B @ b 0 b 1 :::b d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 a 0 a 1 :::a d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 a d c 1 :::c d )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 c d 1 C C C A ; .6.4 A 2 = diag 0 ; 1 ;:::; d ; .6.5 A 3 = tridiag 0 B B B @ b 0 1 b 1 2 :::b d )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 d a 0 a 1 :::a d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 a d c 1 = 1 :::c d )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 c d = d 1 C C C A ; .6.6 forsomenonzero b i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ;c i 2 K 1 i d ,nonzero i 2 K 1 i d ,and a i ; i 2 K 0 i d .Moreover, 0 1 ,..., d aredistinctand c i + a i + b i = 0 0 i d with c 0 = 0, b d = 0. WerstconsiderthecanonicalmodularLeonardtripleswithdiameterone. Lemma2.6.3 [13]Two-by-twomatrices A 1 A 2 A 3 oftheform.6.4{.6.6 with d = 1 formacanonicalmodularLeonardtripleifandonlyif b 0 =+ 2 0 1 / )]TJ/F19 11.9552 Tf 11.61 0 Td [(1 2 c 1 = 1 0 / )]TJ/F19 11.9552 Tf 11.61 0 Td [(1 2 a 0 = 0 b 0 a 1 = 0 c 1 ,and 1 = forsome 0 1 2 K whichsatisfy 6 = 0, 6 = 1, 2 +1 6 = 0,and 0 6 = 1 .We denotethiscanonicalmodularLeonardtripleby MLTO; 0 1 10

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Inthefollowinglemmas,wedisplaythesixfamiliesofcanonicalmodularLeonard tripleswithdiameteratleasttwo. Lemma2.6.4 [13]Let d 2 denoteaninteger.Forany q;;h; 0 2 K whichsatisfy hq 6 =0 q i 6 =1 i d 3 q d + i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 6 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 i d )]TJ/F19 11.9552 Tf 13.105 0 Td [(1 ,and 2 q i 6 =1 i 2 d )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 ,let i = q i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 i d ; i = 0 + h )]TJ/F27 11.9552 Tf 11.955 0 Td [(q i )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 q i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q )]TJ/F28 7.9701 Tf 6.586 0 Td [(i i d ; b 0 = )]TJ/F27 11.9552 Tf 10.494 8.088 Td [(h )]TJ/F27 11.9552 Tf 11.956 0 Td [(q d + 3 q d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q d )]TJ/F27 11.9552 Tf 11.956 0 Td [( ; b i = )]TJ/F27 11.9552 Tf 10.494 8.088 Td [(h )]TJ/F27 11.9552 Tf 11.956 0 Td [(q d )]TJ/F28 7.9701 Tf 6.587 0 Td [(i q )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 q i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + 3 q d + i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q d )]TJ/F28 7.9701 Tf 6.586 0 Td [(i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q i )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2 q 2 i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 i d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; c i = h )]TJ/F27 11.9552 Tf 11.955 0 Td [(q i + q d )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2 q d + i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q d )]TJ/F28 7.9701 Tf 6.586 0 Td [(i +1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(q i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 q 2 i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 i d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; c d = h )]TJ/F27 11.9552 Tf 11.955 0 Td [(q d + q )]TJ/F27 11.9552 Tf 11.955 0 Td [(q d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ; a i = 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b i )]TJ/F27 11.9552 Tf 11.955 0 Td [(c i i d c 0 =0 ;b d =0 : Then A 1 A 2 A 3 from.6.4{.6.6isacanonicalmodularLeonardtripleofdiameterd.WedenotethiscanonicalmodularLeonardtripleby MLTI; d q , h 0 Lemma2.6.5 [13]Let d 2 denoteaninteger.Assume char K is 0 oranoddprime greaterthand.Forany s;h; 0 2 K whichsatisfy h 6 =0 s 6 = i i 2 d ,and 11

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3 s 6 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(2 i d +2 i 2 d +1 ,let i = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 i d ; i = 0 + hi i + s +1 i d ; b 0 = )]TJ/F27 11.9552 Tf 10.494 8.088 Td [(hd s +2 d +4 4 ; b i = h i + s +1 i )]TJ/F27 11.9552 Tf 11.955 0 Td [(d i +3 s +2 d +4 4 i + s +1 i d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; c i = hi i + s + d +1 i )]TJ/F27 11.9552 Tf 11.955 0 Td [(s )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 d )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 4 i + s +1 i d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; c d = )]TJ/F27 11.9552 Tf 10.494 8.088 Td [(hd s +2 4 ; a i = 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b i )]TJ/F27 11.9552 Tf 11.955 0 Td [(c i i d c 0 =0 ;b d =0 : Then A 1 A 2 A 3 from.6.4{.6.6isacanonicalmodularLeonardtripleof diameterd.WedenotethiscanonicalmodularLeonardtripleby MLTII; d s h 0 Lemma2.6.6 [13]Let d 2 denoteaninteger.Assume char K is 0 oraprime greaterthand.Forany ;h; 0 2 K whichsatisfy h 6 =0 6 =1 ,and 1 )]TJ/F27 11.9552 Tf 11.729 0 Td [( + 2 6 =0 let i = i d ; i = 0 + hi i d ; b i = h i )]TJ/F27 11.9552 Tf 11.955 0 Td [(d )]TJ/F27 11.9552 Tf 11.955 0 Td [( + 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 i d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; c i = hi )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 i d ; a i = 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b i )]TJ/F27 11.9552 Tf 11.956 0 Td [(c i i d c 0 =0 ;b d =0 : Then A 1 A 2 A 3 from.6.4{.6.6isacanonicalmodularLeonardtripleof diameterd.WedenotethiscanonicalmodularLeonardtripleby MLTIII; d , h 0 Lemma2.6.7 [13]Let d 2 denoteanoddinteger.Assume char K is 0 oranodd primegreaterthand/2.Forany s;h; 0 2 K whichsatisfy h 6 =0 s 6 =2 i i d foriodd s 6 = i s= 2 6 = d )]TJ/F27 11.9552 Tf 12.929 0 Td [(i +1 i d ,andforieven 3 = 2 s 6 = i + d +2 12

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i d )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 ,let i = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 i i d ; i = 0 + h s )]TJ/F19 11.9552 Tf 11.956 0 Td [(1+ s )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 i )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 i d ; b i = 8 > < > : h i )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 = 2 s + d +2 ifiiseven ; h i )]TJ/F28 7.9701 Tf 6.587 0 Td [(d i )]TJ/F28 7.9701 Tf 6.586 0 Td [(s +1 i )]TJ/F28 7.9701 Tf 6.586 0 Td [(s= 2+1 ifiisodd i d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; c i = 8 > < > : )]TJ/F28 7.9701 Tf 10.494 5.699 Td [(hi i )]TJ/F28 7.9701 Tf 6.586 0 Td [(s + d +1 i )]TJ/F28 7.9701 Tf 6.587 0 Td [(s= 2 ifiiseven ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(h i + s= 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ifiisodd i d ; a i = 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b i )]TJ/F27 11.9552 Tf 11.955 0 Td [(c i i d c 0 =0 ;b d =0 : Then A 1 A 2 A 3 from.6.4{.6.6isacanonicalmodularLeonardtripleof diameterd.WedenotethiscanonicalmodularLeonardtripleby MLTIV; d s h 0 Lemma2.6.8 [13]Let d 2 denoteaneveninteger.Assume char K is 0 oranodd primegreaterthand/2.Forany s;h; 0 2 K whichsatisfy h 6 =0 s 6 =2 i i d foriodd s 6 = i s 6 = d + i +1 3 = 2 s 6 = i + d +1 ,andforieven s= 2 6 = )]TJ/F27 11.9552 Tf 9.299 0 Td [(i + d +2 i d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ,let i = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 i i d ; i = 0 + h s )]TJ/F19 11.9552 Tf 11.955 0 Td [(1+ s )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 i )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 i d ; b i = 8 > < > : h i )]TJ/F27 11.9552 Tf 11.955 0 Td [(d ifiiseven ; h i )]TJ/F25 7.9701 Tf 6.587 0 Td [(3 = 2 s + d +2 i )]TJ/F28 7.9701 Tf 6.586 0 Td [(s +1 i )]TJ/F28 7.9701 Tf 6.586 0 Td [(s= 2+1 ifiisodd i d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; c i = 8 > < > : )]TJ/F28 7.9701 Tf 10.494 5.699 Td [(hi i + s= 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 i )]TJ/F28 7.9701 Tf 6.586 0 Td [(s= 2 ifiiseven ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(h i )]TJ/F27 11.9552 Tf 11.955 0 Td [(s + d +1 ifiisodd i d ; a i = 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b i )]TJ/F27 11.9552 Tf 11.955 0 Td [(c i i d c 0 =0 ;b d =0 : Then A 1 A 2 A 3 from.6.4{.6.6isacanonicalmodularLeonardtripleof diameterd.WedenotethiscanonicalmodularLeonardtripleby MLTV; d s h 0 Lemma2.6.9 [13]Let d 2 denoteaneveninteger.Assume char K is 0 oranodd 13

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primegreaterthand/2.Forany s;h; 0 2 K whichsatisfy h 6 =0 s 6 =2 i i d foriodd s 6 = i s 6 = d + i +1 s= 2 6 = )]TJ/F27 11.9552 Tf 9.299 0 Td [(i + d +1 ,andforieven 3 = 2 s 6 = i + d +2 i d )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 ,let i = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 i +1 i d ; i = 0 + h s )]TJ/F19 11.9552 Tf 11.955 0 Td [(1+ s )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 i )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 i d ; b i = 8 > < > : h i )]TJ/F28 7.9701 Tf 6.586 0 Td [(d i )]TJ/F25 7.9701 Tf 6.587 0 Td [(3 = 2 s + d +2 i )]TJ/F28 7.9701 Tf 6.586 0 Td [(s= 2+1 ifiiseven ; h i )]TJ/F27 11.9552 Tf 11.955 0 Td [(s +1 ifiisodd i d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; c i = 8 > < > : )]TJ/F27 11.9552 Tf 9.299 0 Td [(hi ifiiseven ; )]TJ/F28 7.9701 Tf 10.494 5.698 Td [(hi i )]TJ/F28 7.9701 Tf 6.587 0 Td [(s + d +1 i + s= 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(d )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 i )]TJ/F28 7.9701 Tf 6.586 0 Td [(s= 2 ifiisodd i d ; a i = 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b i )]TJ/F27 11.9552 Tf 11.956 0 Td [(c i i d c 0 =0 ;b d =0 : Then A 1 A 2 A 3 from.6.4{.6.6isacanonicalmodularLeonardtripleof diameterd.WedenotethiscanonicalmodularLeonardtripleby MLTVI; d s h 0 Theorem2.6.10 [13]EverycanonicalmodularLeonardtripleofdiameteratleast twoappearsexactlyonceamongthoseappearinginLemmas2.6.4{2.6.9. Theorem2.6.11 [13]EverymodularLeonardtripleofdiameteratleasttwoisisomorphictosomecanonicalmodularLeonardtriple. 14

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2.7Theautomorphismsandantiautomorphisms InthissectionwestateexplicitlytheantiautomorphismsofLemmas2.3.2and2.4.2, jk i and jk i respectively,usingthefollowingnotation. Denition2.7.1 Let A 1 A 2 A 3 beasinLemma2.6.2.Denediagonalmatrices K N andamatrix R 2 Mat d +1 K asfollows.Theentriesof K N ,and R are K i;i = i Y j =1 b j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 c j ;N i;i = i Y j =1 j i d ; R i;j = i X n =0 Y n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 k =0 i )]TJ/F27 11.9552 Tf 11.955 0 Td [( k j )]TJ/F27 11.9552 Tf 11.955 0 Td [( k Y n k =1 k i;j d ; wherefor1 k d k = b k )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 k )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 Y h =0 k )]TJ/F27 11.9552 Tf 11.955 0 Td [( h = k )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 Y h =0 k )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( h : Theorem2.7.2 [13]Let jk i denotetheantiautomorphismfromLemma2.4.2.With thenotationofDenition2.7.1, 23 1 X = NR )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 N )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X t NR )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 N ; .7.7 13 2 X = KN )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X t KN ; .7.8 12 3 X = KRK )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X t KRK : .7.9 Theorem2.7.3 Let jk i denotetheantiautomorphismfromLemma2.3.2.Withthe notationofDenition2.7.1, 23 1 X = KNN )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X t KNN ; .7.10 13 2 X = KRKN )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 N )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 RK )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X t KRKN )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 N )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 RK ; .7.11 12 3 X = K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X t K : .7.12 15

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Proof. Weobtainresults.7.10and.7.12byapplying N appropriatelyto.7.8 accordingto[12].ByLemma2.4.2,weknowthat 23 1 12 3 = 12 3 13 2 .Thus, 13 2 = 12 3 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 23 1 12 3 .Sincetheantiautomorphisms jk i areinvolutions[13],wehavethat 13 2 X = 12 3 23 1 12 3 X .Then,applying.7.9and.7.10appropriatelyto X yields 13 2 X = 12 3 23 1 12 3 X = 12 3 23 1 K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X t KRK = 12 3 N )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 N )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X t KRK t KNN = K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 N )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 N )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X t KRK t KNN t KRK = K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 N t N t K t K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X t KRKK )]TJ/F28 7.9701 Tf 6.587 0 Td [(t N )]TJ/F28 7.9701 Tf 6.586 0 Td [(t N )]TJ/F28 7.9701 Tf 6.587 0 Td [(t KRK = K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 NNK )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 K )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X t KRKN )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 N )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 RK = KRKN )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 N )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 RK )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X t KRKN )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 N )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 RK : 2 Wenowgiveasimpliedformof N and R foreachofthecanonicalmodular Leonardtripleslistedintheprevioussection.Theseexpressionscanbeobtained directly.Weshallusethenotationalconventionsof[16]concerninghypergeometric series.Thatis,the generalizedhypergeometricseries with r numeratorparameters a 1 ,..., a r and s denominatorparameters b 1 ,..., b s isdenedby r F s a 1 ;a 2 ;:::;a r b 1 ;b 2 ;:::;b s j z = 1 X n =0 a 1 n a 2 n a r n n b 1 n b s n z n ; .7.13 where a n isthe risingfactorial or Pochhammersymbol denedby a n = a a +1 a +2 a + n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 : .7.14 Similarly,the basichypergeometricseries isdenedby r s a 1 ;a 2 ;:::;a r b 1 ;b 2 ;:::;b s j q;z = 1 X n =0 a 1 ; q n a 2 ; q n a r ; q n q ; q n b 1 ; q n b s ; q n )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 n q n 2 1+ s + r z n ; .7.15 16

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where a ; q isthe q-shiftedfactorial denedby a ; q = 8 > < > : 1if n =0 ; )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F27 11.9552 Tf 11.956 0 Td [(aq )]TJ/F27 11.9552 Tf 11.955 0 Td [(aq n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 if n =1 ; 2 ;:::: .7.16 Lemma2.7.4 [13]WiththenotationofDenition2.7.1andLemma2.6.4, N i;i = i q i i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = 2 i d ; R i;j = 4 3 q )]TJ/F29 5.9776 Tf 5.756 0 Td [(i ; 2 q i )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 ;q )]TJ/F29 5.9776 Tf 5.756 0 Td [(j ; 2 q j )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(; )]TJ/F28 7.9701 Tf 6.586 0 Td [( 3 q d )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 ;q )]TJ/F29 5.9776 Tf 5.757 0 Td [(d j q;q i;j d : Lemma2.7.5 [13]WiththenotationofDenition2.7.1andLemma2.6.5, N i;i = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 i i d ; R i;j = 4 F 3 )]TJ/F28 7.9701 Tf 6.586 0 Td [(i;i +1+ s; )]TJ/F28 7.9701 Tf 6.587 0 Td [(j;j +1+ s s= 2+1 ; 3 s= 2+ d +2 ; )]TJ/F28 7.9701 Tf 6.587 0 Td [(d j 1 i;j d : Lemma2.7.6 [13]WiththenotationofDenition2.7.1andLemma2.6.6, N i;i = i i d ; R i;j = 2 F 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(i; )]TJ/F28 7.9701 Tf 6.587 0 Td [(j )]TJ/F28 7.9701 Tf 6.587 0 Td [(d j )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 = )]TJ/F27 11.9552 Tf 11.955 0 Td [( + 2 i;j d : Lemma2.7.7 [13]WiththenotationofDenition2.7.1andLemma2.6.7, N i;i = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 d i= 2 e i d ; k = 8 > < > : )]TJ/F19 11.9552 Tf 9.298 0 Td [(4 h 2 k k )]TJ/F27 11.9552 Tf 11.955 0 Td [(d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ifkiseven ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(4 h 2 k )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 = 2 s + d +1 k )]TJ/F27 11.9552 Tf 11.956 0 Td [(s= 2 ifkisodd k d : 17

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Lemma2.7.8 [13]WiththenotationofDenition2.7.1andLemma2.6.8, N i;i = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 d i= 2 e i d ; k = 8 > < > : )]TJ/F19 11.9552 Tf 9.298 0 Td [(4 h 2 k k )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 = 2 s + d +1 ifkiseven ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(4 h 2 k )]TJ/F27 11.9552 Tf 11.955 0 Td [(d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 k )]TJ/F27 11.9552 Tf 11.955 0 Td [(s= 2 ifkisodd k d : Lemma2.7.9 [13]WiththenotationofDenition2.7.1andLemma2.6.9, N i;i = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 b i= 2 c i d ; k = 8 > < > : )]TJ/F19 11.9552 Tf 9.299 0 Td [(4 h 2 k k )]TJ/F27 11.9552 Tf 11.955 0 Td [(s= 2 ifkiseven ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(4 h 2 k )]TJ/F27 11.9552 Tf 11.955 0 Td [(d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 k )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 = 2 s + d +1 ifkisodd k d : 18

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2.8 Mathematica code:Generalimplementationsetup Weshowsomedetailsofthe Mathematica implementationofcanonicalmodular Leonardtriplesandbegineachsessionbyspecifyingthetypeandthediameter d Type= 3 ; *Takesvalues0,1,2,3,4,5,6,use0fortypeO* d = 2 ; *mustbe1whenType=0* Size= d + 1 ; HerewedenetheentriesofthematriceswhichformacanonicalmodularLeonard triplefromLemma2.6.2oftheTypesI-IIIofLemmas2.6.3{2.6.6. sub=Switch[Type ; 0 ; f nu[ 1 ] nu ; b [ 0 ] 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(nu+nu ^ 2 th[ 0 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(th[ 1 ] = nu )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ^ 2 ; c [ 1 ] nuth[ 1 ] )]TJ/F19 11.9552 Tf 11.956 0 Td [(th[ 0 ] = nu )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ^ 2 g ; 1 ; f nu[i ] nu q i )]TJ/F55 7.9701 Tf 6.587 0 Td [(1 ; th[i ] th[ 0 ]+ h 1 )]TJ/F54 11.9552 Tf 11.955 0 Td [(q i 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(nu 2 q i )]TJ/F55 7.9701 Tf 6.587 0 Td [(1 q )]TJ/F55 7.9701 Tf 6.586 0 Td [(i ; b [ 0 ] !)]TJ/F55 7.9701 Tf 25.77 7.691 Td [(h 1 )]TJ/F55 7.9701 Tf 6.586 0 Td [(q d 1 +nu 3 q d )]TJ/F55 5.9776 Tf 5.756 0 Td [(1 q d 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(nu ; b [i ] !)]TJ/F55 7.9701 Tf 25.77 7.691 Td [(h 1 )]TJ/F55 7.9701 Tf 6.586 0 Td [(q d )]TJ/F55 5.9776 Tf 5.756 0 Td [(i 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(nu 2 q i )]TJ/F55 5.9776 Tf 5.756 0 Td [(1 1 +nu 3 q d + i )]TJ/F55 5.9776 Tf 5.756 0 Td [(1 q d )]TJ/F55 5.9776 Tf 5.756 0 Td [(i 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(nu q i 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(nu 2 q 2i )]TJ/F55 5.9776 Tf 5.756 0 Td [(1 ; c [ d ] h nu 1 )]TJ/F55 7.9701 Tf 6.587 0 Td [(q d 1 +nu q 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(nu q d )]TJ/F55 5.9776 Tf 5.757 0 Td [(1 ; c [i ] h nu 1 )]TJ/F55 7.9701 Tf 6.586 0 Td [(q i 1 +nu q d )]TJ/F55 5.9776 Tf 5.757 0 Td [(i 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(nu 2 q d + i )]TJ/F55 5.9776 Tf 5.756 0 Td [(1 q d )]TJ/F55 5.9776 Tf 5.756 0 Td [(i + 1 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(nu q i )]TJ/F55 5.9776 Tf 5.756 0 Td [(1 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(nu 2 q 2i )]TJ/F55 5.9776 Tf 5.756 0 Td [(1 ; 2 ; f nu[i ] !)]TJ/F54 11.9552 Tf 24.575 0 Td [(1 ; th[i ] th[ 0 ]+ hi i + 1 + s ; b [ 0 ] !)]TJ/F55 7.9701 Tf 25.77 5.699 Td [(hd 3s + 2d + 4 4 ; b [i ] h i + s + 1 i )]TJ/F55 7.9701 Tf 6.587 0 Td [(d 2i + 3s + 2d + 4 4 2i + s + 1 ; c [ d ] )]TJ/F55 7.9701 Tf 6.586 0 Td [(hd s + 2 4 ; c [i ] hi i + s + d + 1 2i )]TJ/F55 7.9701 Tf 6.586 0 Td [(s )]TJ/F55 7.9701 Tf 6.587 0 Td [(2d )]TJ/F55 7.9701 Tf 6.587 0 Td [(2 4 2i + s + 1 o ; 3 ; f nu[i ] nu ; th[i ] th[ 0 ]+ hi ; b [i ] h i )]TJ/F55 7.9701 Tf 6.587 0 Td [(d 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(nu+nu ^ 2 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(nu ^ 2 ; c [i ] hi nu 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(nu ^ 2 o ; ; fg ]; asub= f a [ 0 ] th[ 0 ] )]TJ/F54 11.9552 Tf 11.955 0 Td [(b [ 0 ] ; a [ d ] th[ 0 ] )]TJ/F54 11.9552 Tf 11.955 0 Td [(c [ d ] ; a [j ]> th[ 0 ] )]TJ/F54 11.9552 Tf 11.955 0 Td [(b [ j ] )]TJ/F54 11.9552 Tf 11.955 0 Td [(c [ j ] g ; 19

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WiththesewemaydeneourbasetripletobeginourstudiesfromLemma2.6.2. A [ 1 ]=Table[Which[ i )]TJ/F54 11.9552 Tf 11.955 0 Td [(j === 1 ; c [ i )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ] ; j )]TJ/F54 11.9552 Tf 11.955 0 Td [(i === 1 ; b [ i )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ] ; i === j ; a [ i )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ] ; True ; 0 ] ; f i ; 1 ; Size g ; f j ; 1 ; Size g ]/.asub/.sub; A [ 2 ]=DiagonalMatrix[Table[th[ j )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ] ; f j ; 1 ; Size g ]]/.sub; A [ 3 ]=Table[Which[ i )]TJ/F54 11.9552 Tf 11.955 0 Td [(j === 1 ; c [ i )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ] = nu[ i )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ] ; j )]TJ/F54 11.9552 Tf 11.955 0 Td [(i === 1 ; b [ i )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ]nu[ i ] ; i === j ; a [ i )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ] ; True ; 0 ] ; f i ; 1 ; Size g ; f j ; 1 ; Size g ]/.asub/.sub; Wenowdenethematricies K N ,andtheirinversesfromDenition2.7.1using theformulasfromLemmas2.7.4{2.7.6. KK=DiagonalMatrix[Table[Product[ b [ j )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ] = c [ j ] ; f j ; 1 ; i g ] ; f i ; 0 ; d g ]]/.sub; KI=Inverse[KK]; NN=DiagonalMatrix[Table[ Switch[Type ; 0 ; 1 ; 1 ; nu ^ iq ^ i i )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 = 2 ; 2 ; )]TJ/F54 11.9552 Tf 9.299 0 Td [(1 ^ i ; 3 ; nu ^ i ; ; fg ] ; f i ; 0 ; d g ]]; NI=Inverse[NN]; Wenowdenethematrices R and R )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 fromDenition2.7.1usingtheformulas fromLemmas2.7.4{2.7.6.Weneedafewauxiliaryfunctionstodene R vphi1[g ]:= b [ g )]TJ/F54 11.9552 Tf 11.956 0 Td [(1 ]Product[th[ g ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(th[ h ] ; f h ; 0 ; g )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 g ] = Product[th[ g )]TJ/F54 11.9552 Tf 11.956 0 Td [(1 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(th[ h ] ; f h ; 0 ; g )]TJ/F54 11.9552 Tf 11.955 0 Td [(2 g ]; qsf[a ; q ; n ]:=Product[ 1 )]TJ/F54 11.9552 Tf 11.956 0 Td [(aq ^ k )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ; f k ; 1 ; n g ] 20

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TruncatedBasicHypergeometryPphiQ[A List ; B List ; q ; z ; truncateat ]:= Sum[ z ^ n )]TJ/F54 11.9552 Tf 9.299 0 Td [(q ^ n )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 = 2 ^ n Length[ B ]+ 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(Length[ A ]Apply[Times ; Map[Function[ x ; qsf[ x ; q ; n ]] ; A ]] = qsf[ q ; q ; n ]Apply[Times ; Map[Function[ x ; qsf[ x ; q ; n ]] ; B ]] ; f n ; 0 ; truncateat g ] RR=Simplify[Table[Switch[Type ; 0 ; Sum[Product[th[ i ] )]TJ/F19 11.9552 Tf 11.956 0 Td [(th[ k ]th[ j ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(th[ k ] ; f k ; 0 ; n )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 g ] = Product[vphi1[ k ] ; f k ; 1 ; n g ] ; f n ; 0 ; i g ]/.sub ; 1 ; TruncatedBasicHypergeometryPphiQ[ f q ^ )]TJ/F54 11.9552 Tf 9.298 0 Td [(i ; nu ^ 2q ^ i )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ; q ^ )]TJ/F54 11.9552 Tf 9.299 0 Td [(j ; nu ^ 2q ^ j )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 g ; f)]TJ/F19 11.9552 Tf 15.276 0 Td [(nu ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(nu ^ 3q ^ d )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ; q ^ )]TJ/F54 11.9552 Tf 9.299 0 Td [(d g ; q ; q ; d ]/.sub ; 2 ; HypergeometricPFQ[ f)]TJ/F54 11.9552 Tf 15.276 0 Td [(i ; i + 1 + s ; )]TJ/F54 11.9552 Tf 9.299 0 Td [(j ; j + 1 + s g ; f s = 2 + 1 ; 3s = 2 + d + 1 + 1 ; )]TJ/F54 11.9552 Tf 9.299 0 Td [(d g ; 1 ] ; 3 ; HypergeometricPFQ[ f)]TJ/F54 11.9552 Tf 15.276 0 Td [(i ; )]TJ/F54 11.9552 Tf 9.299 0 Td [(j g ; f)]TJ/F54 11.9552 Tf 15.276 0 Td [(d g ; 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(nu ^ 2 = 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(nu+nu ^ 2 ] ; ; fg ] ; f i ; 0 ; d g ; f j ; 0 ; d g ]]; RRI=Switch[Type ; 0 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(nu/ )]TJ/F54 11.9552 Tf 9.298 0 Td [(1 +nu 2 ; 1 ; qsf[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(nu ; q ; d ] = qsf[nu ^ 2 ; q ; d ] ^ 2 )]TJ/F19 11.9552 Tf 9.298 0 Td [(nu ^ d ; 2 ; )]TJ/F54 11.9552 Tf 9.299 0 Td [(1 ^ d 2 ^ )]TJ/F54 11.9552 Tf 11.956 0 Td [(d Pochhammer[ s = 2 + 1 +Ceiling[ d = 2 ] ; Floor[ d = 2 ]] = Pochhammer[ s + 3 = 2 ; Floor[ d = 2 ]] ^ 2 ; 3 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(nu ^ d nu )]TJ/F54 11.9552 Tf 11.955 0 Td [(1 ^ )]TJ/F54 11.9552 Tf 11.955 0 Td [(d ^ 2 ; ; fg ]KK : RR : KK; WenowimplementtheantiautomorphismsofLemma2.3.2byusingTheorem 2.7.3. alpha[ 1 ; x ]:=NI : NI : KI : Transpose[ x ] : KK : NN : NN alpha[ 2 ; x ]:=KI : RRI : NN : NN : KI : RRI : KI : Transpose[ x ] : KK : RR : KK : NI : NI : RR : KK alpha[ 3 ; x ]:=KI : Transpose[ x ] : KK 21

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3 -orbitsofLeonardtriples SinceanytwoelementsofaLeonardtripleformaLeonardpairLemma2.3.2,wemay applytheantiautomorphismxinganyofthepairstothethirdelementtoproducea possiblynewLeonardtripleLemma2.3.1.Inthischapterweintroducethenotion ofan -orbittocaptureallLeonardtripleswhicharisebyiteratingthisprocess. 3.1Validsequencesofantiautomorphisms Inthissectionweintroducethenotionofavalidsequenceofantiautomorphisms.We willformmorepossiblynewLeonardtriplesbycomposingtheelementsofthese sequencesandapplyingthesecompositionstoelementsofaLeonardtriple.Priorto introducingthisnotion,weshowsomebasicpropertiesofantiautomorphisms. Lemma3.1.1 Let A 1 A 2 A 3 beaLeonardtriple.Fixapermutation ijk of 123 ,and let A i 0 = jk i A i ,where jk i isasinLemma2.3.2.Then jk i = jk i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ; .1.1 jk i = kj i ; .1.2 jk i = jk i 0 : .1.3 Proof. Result.1.1followsfromLemma2.3.1since jk i isaninvolution.Result .1.2followsfromthefactthatxing A j and A k isthesameasxing A k and A j Result.1.3followsfromthefactthatanantiautomorphismxing A j and A k sends A i to A i 0 andviceversasinceantiautomorphismsareunique. 2 Denition3.1.2 Let A 1 A 2 A 3 beaLeonardtriple.Wesaythat 23 1 13 2 12 3 isa validsequence oflength1ofantiautomorphismsofEnd V for A 1 A 2 A 3 .Wesay 22

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thatasequence 1 2 ;:::; ` ofantiautomorphismsofEnd V isavalidsequenceof length ` for A 1 A 2 A 3 when 1 2 ;:::; ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 isavalidsequencefor A 1 A 2 A 3 ` 6 = ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 and ` xesatleasttwoof ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ::: 2 1 A 1 for i 2f 1 ; 2 ; 3 g .Wereferto as the composition ofthevalidsequence.Wereferto j = j j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ::: 2 1 j ` asthe j th partialcomposition ofavalidsequence. Notethatdistinctvalidsequencesmayhavethesamecompositions. Lemma3.1.3 Let and becompositionsofvalidsequencesasinDenition3.1.2. Thenthefollowingareequivalent: i = ii A i = A i for i 2f 1 ; 2 ; 3 g Proof. ThisresultfollowsfromthefactthateachLeonardpairon V generates End V Lemma2.2.3. 2 WerenethisresultinCorollary4.1.6. Denition3.1.4 Let 1 2 ;:::; ` beavalidsequenceforaLeonardtriple A 1 A 2 A 3 .Then i xesatleasttwoof i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A 1 i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 A 2 i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A 3 .Let p i q i 2f 1 ; 2 ; 3 g be suchthat i xes i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A p i and i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A q i .Let r i 2f 1 ; 2 ; 3 gnf p i ;q i g .Wealsodenote thisvalidsequencewiththedata p 1 ;q 1 ; r 1 p 2 ;q 2 ; r 2 ::: p ` ;q ` ; r ` Observethat p i ;q i ; r i and q i ;p i ; r i describethesameantiautomorphism.If weapplyavalidsequenceasinDenition3.1.4toonly A 1 A 2 A 3 ,thesequence isuniquelydenedbythevalidsequence[ r 1 ;r 2 ;r 3 ;:::;r ` ].Notethatif i alsoxes i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 A r i ,then i canbedescribedintheoneofthefollowing6waysallpermutations of p i q i r i : p i ;q i ; r i q i ;p i ; r i p i ;r i ; q i q i ;r i ; q i r i ;p i ; q i r i ;q i ; p i .Thus6 permutationsof123nametheantiautomorphisms,butonlyhalfasmanyaredistinct sincetheycorrespondtoanyxedpairbeingunordered.Inavalidsequencewe cannothavethesameantiautomorphismappeartwiceconsecutivelysothereareonly twowaystocontinueateachstage. Lemma3.1.5 Thereareatmost 3 2 ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 validsequencesofantiautomorphismsof length ` ` 1 foreachLeonardtriple. 23

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Proof. ThisresultfollowsfromthefactthataLeonardpaircanappearinatmost twoLeonardtriplesandfromdiscussionabove[12]. 2 24

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3.2 -orbitsfromvalidsequences Inthissectionweusethenotionofvalidsequencestobuild -orbitsofLeonardtriples. Denition3.2.1 Let A 1 A 2 A 3 beaLeonardtriple.The -orbit istheimageof A 1 A 2 A 3 underallcompositionsofvalidsequences.Thatistosay,the -orbitof A 1 A 2 A 3 isthesetofLeonardtriplessuchthatthereexistsasequenceofLeonard triples A 1 = A 1 A 2 = A 2 A 3 = A 3 ; A 1 A 2 A 3 ; . A k 1 = B 1 A k 2 = B 2 A k 3 = B 3 ; suchthat A i +1 1 = A i 1 A i +1 2 = A i 2 A i +1 3 = A i 3 ,forsomeantiautomorphism whichxesatleasttwoof A i 1 A i 2 A i 3 Lemma3.2.2 If A 1 A 2 A 3 isinthe -orbitof B 1 B 2 B 3 ,then B 1 B 2 B 3 isin the -orbitof A 1 A 2 A 3 Proof. Suppose A 1 A 2 A 3 isinthe -orbitofsomeLeonardtriple B 1 B 2 B 3 .Then byDenition3.2.1,thereexistsasequenceofantiautomorphismswhichwhenapplied to A 1 A 2 A 3 yield B 1 B 2 B 3 andviceversa. 2 Lemma3.2.3 The -orbitof A 1 A 2 A 3 istheintersectionofallsetsofLeonard triplesforwhichthefollowinghold: i Thesetcontainstheoriginaltriple A 1 A 2 A 3 ii IftwoLeonardtriplesdierinoneplace,theneitherbothareinthesetorneither is. Proof. OnecontainmentfollowsdirectlyfromDenition3.2.1sincethe -orbitmust containtheoriginaltripleandsincethe -orbitisformedbyapplyingtheappropriate antiautomorphismwhichxesatleasttwoelementsoftheprevioustriple,soeither bothtriplesareinthe -orbitorneitherare.Thus,theintersectionofallsetsof 25

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LeonardtriplessuchthatPartsiandiihold,iscontainedinthe -orbitof A 1 A 2 A 3 .Fortheotherdirection,let A i 1 A i 2 A i 3 beaLeonardtripleinthe -orbitof A 1 A 2 A 3 .Theneither i =0and A i 1 = A 1 A i 2 = A 2 A i 3 = A 3 ,theoriginaltriple, orthereexistssomeantiautomorphism whichxesatleasttwoof A i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 1 A i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 2 A i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 3 .Thus,if i 6 =0,thesetwoLeonardtripleswhichdierinoneplacearebothin the -orbitoftheoriginaltriple.Thus,the -orbitof A 1 A 2 A 3 iscontainedinthe intersectionofallsetsofLeonardtriplessuchthatPartsiandiihold.Hence,we havethedesiredequality. 2 Denition3.2.4 Let bethecompositionofavalidsequenceoflength ` .Wepartitionthe -orbitof A 1 A 2 A 3 into stages by ` = f A 1 ; A 2 ; A 3 gn S ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 j =1 j Wereferto ` asthe ` th stage ofthe -orbitgrowthof A 1 A 2 A 3 Theorem3.2.5 Let p 1 ;q 1 ; r 1 p 2 ;q 2 ; r 2 ::: p ` ;q ` ; r ` beavalidsequenceofantiautomorphismsforaLeonardtriple A 1 A 2 A 3 .Let bethecompositionofthisvalid sequence.Then 9 2f 23 1 ; 13 2 ; 12 3 g andsome whichisthecompositionof ` )]TJ/F19 11.9552 Tf 12.329 0 Td [(1 elementsfrom f 23 1 ; 13 2 ; 12 3 g suchthat = )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 .Infact,wemaytake = p ` )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 q ` )]TJ/F26 5.9776 Tf 5.757 0 Td [(1 r ` )]TJ/F26 5.9776 Tf 5.757 0 Td [(1 p ` )]TJ/F26 5.9776 Tf 5.756 0 Td [(2 q ` )]TJ/F26 5.9776 Tf 5.756 0 Td [(2 r ` )]TJ/F26 5.9776 Tf 5.756 0 Td [(2 ::: p 1 q 1 r 1 ; .2.4 = p ` q ` r ` : .2.5 Proof. Proceedbyinduction,beginningwith ` =1.Let 1 beavalidsequencefora modularLeonardtriple A 1 A 2 A 3 .ThenbyDenition3.1.2,wehave = 1 xesat leasttwoof A 1 A 2 A 3 .Thus, 1 2f 23 1 ; 13 2 ; 12 3 g .Take = 1 and = id .Then = 1 = )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 andtheresultholds.Nowlet = ` ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ::: 2 1 foranyvalidsequence 1 2 ;:::; ` .Assumethereissome 2f 23 1 ; 13 2 ; 12 3 g andsome oflength ` )]TJ/F19 11.9552 Tf 11.111 0 Td [(1 suchthatthe = )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 .Take = andlet 2f 23 1 ; 13 2 ; 12 3 g ,where 6 = .Then, )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 .Thenbyinduction hypothesis,wehavethat )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 xesatleasttwoelementsof ` ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ::: 2 1 A i for i 2f 1 ; 2 ; 3 g .Hence,foravalidsequence 1 2 ;:::; ` ` +1 ,wecannda 2f 23 1 ; 13 2 ; 12 3 g anda oflength ` suchthat = ` +1 ` ::: 2 1 = )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 Moreover, and areoftheappropriateformsin.2.4and.2.5respectively. Hencetheresultfollowsfrominduction. 2 26

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Lemma3.2.6 Let p 1 ;q 1 ; r 1 p 2 ;q 2 ; r 2 ::: p ` ;q ` ; r ` beavalidsequenceofantiautomorphismsforaLeonardtriple A 1 A 2 A 3 .Say bethecompositionofthisvalid sequence.Say xestwoof A 1 A 2 A 3 ,say A p ` +1 and A q ` +1 and let r i 2f 1 ; 2 ; 3 gnf p i ;q i g .Then p 1 ;q 1 ; r 1 p 2 ;q 2 ; r 2 ::: p ` ;q ` ; r ` p ` +1 ;q ` +1 ; r ` +1 isa validsequenceanditscompositionisequalto p ` q ` r ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 Proof. Observethat )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 mustxatleasttwoof ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ::: 2 1 A i for i 2f 1 ; 2 ; 3 g ByDenition3.1.2, = )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 isthecompositionofavalidsequenceoflength ` 2 Table3.1liststherstthreestagesofpossible -orbitgrowthofaLeonardtriple. Table3.1:Beginningstagesof -orbitgrowth p i ; q i ; r i = )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 FixedElements Result ; 3;1 23 1 A 2 A 3 A 1 7! 23 1 A 1 ; 3;2 13 2 A 1 A 3 A 2 7! 13 2 A 2 ; 2;3 12 3 A 1 A 2 A 3 7! 12 3 A 3 ; 3;1 ; 3;2 23 1 13 2 23 1 A 3 23 1 A 1 A 2 7! 23 1 13 2 A 2 ; 3;1 ; 2;3 23 1 12 3 23 1 A 2 23 1 A 1 A 3 7! 23 1 12 3 A 3 ; 3;2 ; 3;1 13 2 23 1 13 2 A 3 13 2 A 2 A 1 7! 13 2 23 1 A 1 ; 3;2 ; 2;3 13 2 12 3 13 2 A 1 13 2 A 2 A 3 7! 13 2 12 3 A 3 ; 2;3 ; 3;1 12 3 23 1 12 3 A 2 12 3 A 3 A 1 7! 12 3 23 1 A 1 ; 2;3 ; 3;2 12 3 13 2 12 3 A 1 12 3 A 3 A 2 7! 12 3 13 2 A 2 ; 3;1 ; 3;2 ; 3;1 23 1 13 2 23 1 13 2 23 1 A 3 23 1 13 2 A 2 23 1 A 1 7! 23 1 13 2 23 1 A 1 ; 3;1 ; 3;2 ; 2;3 23 1 13 2 12 3 13 2 23 1 23 1 A 1 23 1 13 2 A 2 A 3 7! 23 1 13 2 12 3 A 3 ; 3;1 ; 2;3 ; 3;1 23 1 12 3 23 1 12 3 23 1 A 2 23 1 12 3 A 3 23 1 A 1 7! 23 1 12 3 23 1 A 1 ; 3;1 ; 2;3 ; 3;2 23 1 12 3 13 2 12 3 23 1 23 1 A 1 23 1 12 3 A 3 A 2 7! 23 1 12 3 13 2 A 2 ; 3;2 ; 3;1 ; 3;2 13 2 23 1 13 2 23 1 13 2 A 3 13 2 23 1 A 1 13 2 A 2 7! 13 2 23 1 13 2 A 2 ; 3;2 ; 3;1 ; 2;3 13 2 23 1 12 3 23 1 13 2 13 2 A 2 13 2 23 1 A 1 A 3 7! 13 2 23 1 12 3 A 3 ; 3;2 ; 2;3 ; 3;1 13 2 12 3 23 1 12 3 13 2 13 2 A 2 13 2 12 3 A 3 A 1 7! 13 2 12 3 23 1 A 1 ; 3;2 ; 2;3 ; 3;2 13 2 12 3 13 2 12 3 13 2 A 1 13 2 12 3 A 3 13 2 A 2 7! 13 2 12 3 13 2 A 2 ; 2;3 ; 3;1 ; 3;2 12 3 23 1 13 2 23 1 12 3 12 3 A 3 12 3 23 1 A 1 A 2 7! 12 3 23 1 13 2 A 2 ; 2;3 ; 3;1 ; 2;3 12 3 23 1 12 3 23 1 12 3 A 2 12 3 23 1 A 1 12 3 A 3 7! 12 3 23 1 12 3 A 3 ; 2;3 ; 3;2 ; 3;1 12 3 13 2 23 1 13 2 12 3 12 3 A 3 12 3 13 2 A 2 A 1 7! 12 3 13 2 23 1 A 1 ; 2;3 ; 3;2 ; 2;3 12 3 13 2 12 3 13 2 12 3 A 1 12 3 13 2 A 2 12 3 A 3 7! 12 3 13 2 12 3 A 3 27

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3.3 Mathematica code:ApplyingValidSequences Wenowincludesome Mathematica codethatshowshowwewillusevalidsequences describedinthischapterandapplythemtoatriple. InlightofthecommentsfollowingDenition3.1.4wespecifyavalidsequenceof length ` asalistof ` numbersfrom f 1 ; 2 ; 3 g suchthatnoconsecutivenumbersare equal.Inordertocomputethecomposition,weneedtoexpressitintermsofthe localantiautomorphisms 12 3 13 2 23 1 .WedosobyusingTheorem3.2.5. RelativeToLocal[ fg ]:= fg RelativeToLocal[valseq ]:=Join[valseq ; Rest[Reverse[valseq]]] Applythecompositionofalistofantiautomorphismsfromthebasicthreetoa giventripleofmatrices. ApplyLocalAntiAut[ fg ; threetuple ]:=threetuple ApplyLocalAntiAut[aaseq ; threetuple ]:=Map[alpha[First[aaseq] ; #]& ; ApplyLocalAntiAut[Rest[aaseq] ; threetuple]] ApplyavalidsequencetothestartingmodularLeonardtriple.Needtobuildin factthatonlyonemovesateachstage. ApplyValidSequence[ fg ]:=ModularLeonardTriple ApplyValidSequence[valseq ]:=ApplyLocalAntiAut[RelativeToLocal[valseq] ; ModularLeonardTriple] 28

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3.4Thecomplexofan -orbit Inthissectionweusethenotionofasimplicialcomplextogiveageometricinterpretationofthe -orbitofaLeonardtriple. Denition3.4.1 An n-simplex istheconvexhullofasetof n +1anelyindependentpointsinsomeEuclideanspaceofdimension n orhigher.Forexample,apoint isa0-simplex,alinesegmentisa1-simplex,andatriangleisa2-simplex. Weshallidentify0-simpliceswiththeelementsofeachLeonardtripleinan -orbit, 1-simpliceswitheachLeonardpairinan -orbit,and2-simpliceswitheachLeonard tripleinan -orbit. Denition3.4.2 Theconvexhullofanynonemptysubsetofthe n +1pointsthat denean n -simplexiscalleda face ofthesimplex.Observethatfacesaresimplices themselves. Denition3.4.3 A simplicialcomplex isasetofsimplicesthatsatisesthefollowingconditions: i Anyfaceofasimplexfromisalsoin. ii Theintersectionofanytwosimplices s 1 ;s 2 2 isafaceofboth s 1 and s 2 Weconsiderwhichsimplicialcomplexesmayarisefroman -orbitofaLeonardtriple. Lemma3.4.4 The -orbitofaLeonardtripleformsasimplicialcomplex withthe followingproperties: i Thecomplexisnotdisconnectedbytheremovalofasingle0-simplex. ii Eachedgeofthecomplexisinatmosttwofaces. Proof. TheconditionsofDenition3.4.3aresatisedsinceanypointinatriangle isalsoanelementofthecomplexofDenition3.4.3andsincetheintersectionof anytwotrianglesfromthecomplexisanedgeofbothtrianglesofDenition3.4.3i. Moreover,sinceeachnewtriangularfaceofthecomplexisobtainedbyapplyingantiautomorphismswhichxtwoelementsofthepreviousface,itfollowsthatremoving 29

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asinglevertexwillnotdisconnectthecomplex.SinceeveryLeonardpaircanappear inatmost2Leonardtriples,andsinceeachedgerepresentsaLeonardpairandeach facerepresentsaLeonardtriple,theneachedgeofcomplexisinatmosttwofaces. Thus,theresultholds. 2 Denition3.4.5 Let bethesimplicialcomplexofaLeonardtriple A 1 A 2 A 3 Wereferto asthe complex ofaLeonardtripleandspecifythecomplexusingthe notation A 1 ;A 2 ;A 3 Lemma3.4.6 Leonardtriplesinthesame -orbithavethesamecomplex. Proof. IfaLeonardtriple B 1 B 2 B 3 isinthesame -orbitasanotherLeonardtriple A 1 A 2 A 3 ,thentheremustexistsomesequenceofantiautomorphismswhichwhen appliedto A 1 A 2 A 3 yields B 1 B 2 B 3 andviceversa.Therefore,thesamefaces thatintersectinthecomplexcontaining B 1 B 2 B 3 mustintersectinthecomplex containing A 1 A 2 A 3 .Thusthecomplexof B 1 B 2 B 3 andthecomplexof A 1 A 2 A 3 mustbethesameandtheresultholds. 2 Lemma3.4.7 ThecomplexofaLeonardtripleisthesmallestsetoflineartransformationscontainingthegivenLeonardtripleandisclosedundertheantiautomorphisms ofEnd V xinganytwoelementsofthecomplexwhichformaLeonardpair. Proof. ProoffollowsdirectlyfromLemmas3.2.3and3.4.6. 2 30

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3.5Symmetryofan -orbit'scomplex Inthissection,wewillshowthateach -orbitgivesrisetoacomplex .Wewillalso showthateach iseitheratriangleoreveryedgeof isinexactly2faces. Denition3.5.1 An automorphism ofacomplex isabijection : which respectscontainment.Thatis,for x;y 2 ,if x y ,then x y .Observethat isdeterminedbyitsbehavioron ,sowemayidentifyitwithapermutationofthe 0-simplices. Theorem3.5.2 TheantiautomorphismsfromLemmas2.3.2induceanautomorphismofthecomplex ofeachLeonardtriple. Proof. Fix ijk tobeapermutationof123foraLeonardtriple A 1 A 2 A 3 .From Lemma2.3.2,each jk i allowsustoxanytwoverticesandeithergettoanotherface ofthecomplexorstayinthesameface.Thusperformingasequenceofantiautomorphismsresultsinpermutingthe0-simplicesofthecomplex.Hence,wehavethe necessaryautomorphismsmappinganyfaceofthecomplextoanyotherfaceofthe complex 2 Byconstruction,wecanmapany2-simplextoanyother2-simplex,however,we maynothavemuchcontrolofthemappingof0-simplices. Denition3.5.3 Let bethecomplexofaLeonardtriple.Let v 2 bea0-simplex. Bythe degree of v deg 1 v ,wemeanthenumberof1-simplicescontaining v .Wesay acomplexis regular ifeveryvertexhasthesamedegree. Lemma3.5.4 Let bethecomplexofaLeonardtriple.Let jk i v betheimageofa vertex v 2 undersomeantiautomorphism jk i .Then deg 1 v = deg 1 jk i v Proof. TheresultfollowsclearlyfromTheorem3.5.2. 2 ByLemma3.5.4wehaveahighdegreeofsymmetryineachcomplex.Thishigh degreeofsymmetryresultsinthefollowingLemma. Denition3.5.5 Wecallthegraphwhoseverticesandedgesarethoseofthecomplex ofaLeonardtriple,the graphoftheLeonardtriple 31

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Denition3.5.6 An edge-transitive graphisagraph G suchthat,givenanytwo edges e 1 and e 2 of G ,thereisanautomorphismof G thatmaps e 1 to e 2 .Inother words,agraphisedge-transitiveifitsautomorphismgroupactstransitivelyuponits edges. Lemma3.5.7 Thegraph G ofaLeonardtripleisedge-transitive. Proof. Sincethereexistsanautomorphismthatmapseveryfaceofacomplex to anyotherfaceof byLemma3.5.4,thenitfollowsthateveryedgeofthegraph G canbemappedtoanyotheredgeof G .ThusbyDenition3.5.6,wehavethat G is edge-transitive. 2 Denition3.5.8 A distance-transitive graphisagraph G suchthat,givenanytwo vertices v and w atanydistance i ,andanyothertwovertices x and y atthesame distance,thereisanautomorphismof G thatcarries v to x and w to y Denition3.5.9 A distance-regular graphisagraph G suchthat,givenanytwo vertices v and w atanydistance i ,thenumberofverticesadjacentto w andat distance j from v dependsonlyon i and j andthedistancebetween w and v ThenextexampleshowsthatgraphsofLeonardtriplesarenotnecessarilydistancetransitiveordistance-regular. Example3.5.10 WeconsiderthegraphofaLeonardtriplewhosecomplexisa7regulartesselationofthehyperbolicplaneConsiderthegraphofthistripleshown inFigure3.1below.Notethatalthough v and w arethesamedistanceapartas x and y ,noautomorphismexistsmapping v to x and w to y .Thus,theagraphof aLeonardtripleisnotnecessarilydistance-transitivedespitebeingedge-transitive. Also,notethatthedistancebetween w and v is2andthedistancebetween x and v is2,however w and v haveonly1adjacentvertexincommon,while x and v have 2adjacentverticesincommon.Hence,byDenition3.5.9,thegraphofaLeonard tripleisnotnecessarilydistance-regular. Problem3.5.11 Ifthe -orbitofaLeonardtripleisnite,mustthegraphofthe Leonardtriplebedistance-transitiveordistance-regular? 32

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Figure3.1:Counterexampleofdistance-transitivity/regularity Theorem3.5.12 TheantiautomorphismsfromLemma2.3.2andTheorem2.7.2induceanautomorphismofthecomplex ofeachmodularLeonardtriple. Proof. Fix ijk tobeapermutationof123foraLeonardtriple A 1 A 2 A 3 .From Lemma2.3.2,each jk i allowsustoxanytwoverticesandeithergettoanother faceofthecomplexorstayinthesameface.FromTheorem2.7.2,each jk i allows ustoxonevertexofatriangleandswaptheothertwovertices.Thusperforminga sequenceofantiautomorphismsresultsinpermutingthe0-simplicesofthecomplex. Hence,wecanmapanymodularLeonardtripletoanyothermodularLeonardtriple inthesamecomplex 2 Denition3.5.13 An arc-transitive graphisagraph G suchthatthereexistsan automorphismmapping v 1 v 2 to v 0 1 v 0 2 foralledges v 1 v 2 v 0 1 v 0 2 2 G Lemma3.5.14 ThegraphofamodularLeonardtripleisarc-transitive. Proof. If A 1 A 2 A 3 isamodularLeonardtriple,thenwecanmapanyedgetoany otheredgeinitsgraphsincewehavethe S 3 actionofTheorem2.7.2andtheipping actionofTheorem2.7.3. 2 33

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Lemma3.5.15 Let G bethegraphofaLeonardtriple.Supposethereexistsan automorphismof G inducedbytheantiautomorphismsofEnd V mapping v 1 v 2 to v 0 1 v 0 2 foralledges v 1 v 2 v 0 1 v 0 2 2 G .Thenthetriplemustbemodular. Proof. Sincewecanmapanyedgetoanyedgebyanautomorphisminducedbythe antiautomorphismsofEnd V ,itfollowsthattheremustexistanantiautomorphism ofEnd V whichxesonememberofthetripleandswapstheothertwomembers. Thus,thetripleismodularbyDenition2.4.1. 2 Problem3.5.16 SupposethegraphofaLeonardtripleisregularandthatnoantiautomorphisminducesanyoftheautomorphismsofthegraph.Isthegrapharctransitive? Theorem3.5.17 Let bethecomplexofamodularLeonardtriple.Then isregular inthesenseofDenition3.5.3. Proof. ByLemma3.5.14,sincethegraph G ofamodularLeonardtripleisarctransitive,thereexistsanautomorphismmapping v 1 v 2 to v 0 1 v 0 2 foralledges v 1 v 2 v 0 1 v 0 2 2 G .Thus,everyvertexof hasthesamedegreeandtheresultfollows. 2 Problem3.5.18 ObservethattheLeonardtriplesofan -orbitareeitherisomorphic oranti-isomorphic.IftwomodularLeonardtriplesisomorphic,aretheynecessarily inthesame -orbit? SeeFigure3.2forhowwebegintoformacomplexusingantiautomorphisms. 34

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Figure3.2:AntiautomorphismactiononLeonardtriples 35

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3.6Vertex -orbitgrowth InthissectionwexavertexofatriangleandmovearoundthatvertexasinFigure 3.3.Weconsiderthecasethatthisvertexisinanitenumberofedges. Figure3.3:Vertex -orbitgrowth Lemma3.6.1 Let A 1 A 2 A 3 beaLeonardtripleandlet ijk beapermutationof 123 .Thentherearetwovalidsequences 1 2 ;:::; ` suchthat h A i = A i 8 h ,for somexed i 2f 1 ; 2 ; 3 g ,namelythosewiththefollowing p i ;q i ; r i ofTheorem3.2.5: i;j ; k i;k ; j i;j ; k ::: i;k ; j i;j ; k i;k ; j :::: Proof. Let ijk beapermutationof123.SinceforeachLeonardtriplewehave3 unorderedpairsandforeachofthesepairsthereisanantiautomorphismwhichxes bothelements,wecanstartwithaLeonardtripleandxthepair A i A j orthepair A i A k .Continuinginthisway,wex A i andreturnto A j and A k byapplyingavalida sequenceoftheform i;j ; k i;k ; j i;j ; k ::: ,oroftheform i;k ; j i;j ; k i;k ; j ::: 2 Denition3.6.2 Let A 1 A 2 A 3 beaLeonardtriple.For i 2f 1 ; 2 ; 3 g ,wesay A i is vertex-nite withrespectto A 1 A 2 A 3 ifthefollowingsetisnite: f A 1 ; A 2 ; A 3 j isthecompositionofavalidsequenceasinLemma3.6.1 g 36

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Denition3.6.3 Bythe degreeof A i withrespectto A 1 A 2 A 3 wemeantheminimal positivelength ` ofanyvalidsequencewithcomposition suchthat A 1 = A 1 A 2 = A 2 A 3 = A 3 .Wesaythedegreeis innite ifthereisnosuch ` Theorem3.6.4 Let A 1 A 2 A 3 beaLeonardtriple. i Ifthe -orbitisnite,theneveryvertexofthecomplexhasnitedegree. ii Ifsomevertexofthecomplexisnotvertex-nite,thenthe -orbitisinnite. Proof. Supposethe -orbitisnite.Proceedtogrowtheorbitbyxingavertexofatriangleandmovingaroundit.Sincethe -orbitisnite,eventuallysome triplemustberepeated.Thentheremustexistsomepositivelength ` ofanyvalid sequencewithcomposition suchthat A 1 = A 1 A 2 = A 2 A 3 = A 3 .Thus, f A 1 ; A 2 ; A 3 g isniteas runsoverthecompositionofallvalidsequences 1 2 ;:::; m suchthat h A i = A i for1 h m )]TJ/F19 11.9552 Tf 12.157 0 Td [(1.Sowehavethateach A i isvertex-nitewithrespect A 1 A 2 A 3 asinDenition3.6.2andPartiholds.Part iiisjustthecontrapositiveofParti. 2 Lemma3.6.5 Let A 1 A 2 A 3 beaLeonardtriple.If A i isvertex-nitewithrespect A 1 A 2 A 3 withdegree ` ,then A i isvertex-nitewithrespectto A 1 A 2 A 3 withdegree ` ,foranycomposition ofvalidsequences. Proof. Since A i isvertex-nitewithrespect A 1 A 2 A 3 withdegree ` ,wehavethat A 1 = A 1 A 2 = A 2 A 3 = A 3 forsomecomposition ofavalidsequenceof length ` .Thus, A i = A i isvertex-nitewithrespectto A 1 = A 1 A 2 = A 2 A 3 = A 3 withdegree ` andtheresultfollows. 2 ObservethatbyLemma3.6.5,thegraphofaLeonardtripleis tri-regular inthe sensethatwecanpartitiontheverticesintothreesetswhereverticesinthesameset havethesamedegree.Anexampleofatri-regulargraphisthecompletetripartite graph K 3 ; 2 ; 3 inFigure3.4wherevertices a b and c eachhavedegree5,vertices d and e eachhavedegree6,andvertices f g ,and h eachhavedegree5.Notethatwith modularityextendedtoLeonardtriples,wehavethestrongernotionofregularity. 37

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Figure3.4:Completetripartitegraph K 3 ; 2 ; 3 Lemma3.6.6 Let A 1 A 2 A 3 beamodularLeonardtriple.Assumethateach A i isvertex-nitewithrespectto A 1 A 2 A 3 .Theneach A i hasthesamedegreewith respectto A 1 A 2 A 3 Proof. Since A 1 A 2 A 3 isamodularLeonardtriple,thereexistsanantiautomorphism whichxesoneelementofthetripleandswapstheremainingelementsThus,wehave S 3 actiononthetriple.Itfollowsthateachvertexmustappearinthesamenumberof triples,hencetheminimalpositivelength ` ofanyvalidsequencewithcomposition suchthat A i = A i ,mustbethesameforeach A 1 A 2 ,and A 3 .Thus,byDenition 3.6.3,each A i hasthesamedegreewithrespecttotheoriginaltriple. 2 Problem3.6.7 AssumethateveryvertexinthecomplexofaLeonardtripleis vertex-nite.If2verticesofatriplehaveequaldegree,doesthethirdvertexhavethe samedegree? Wedene Mathematica codeforperformingthevertexrotationsfromthissection. AlternatingSeq[xed ; steps ; rst ]:=Module[ f other=Select[ f 1 ; 2 ; 3 g ; And[# 6 =rst ; # 6 =xed]&][[ 1 ]] g ; Table[If[OddQ[ i ] ; rst ; other] ; f i ; 1 ; steps g ]] VertexRotate[xed ; steps ; rst ]:= ApplyValidSequence[AlternatingSeq[xed ; steps ; rst]] 38

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3.7Channel -orbitgrowth Inthissectionwedonotxoneelementbutratherfollowastraightchannelpathof antiautomorphismsfromatripleasinFigure3.5.Werefertothistypeofgrowthas channel -orbitgrowth.Wegiveanitenessconditionthatinvolveschannel -orbit growth.Inaddition,wegiveanecessaryconditionfornite -orbitsandasucient conditionforinnite -orbits. Figure3.5:Channel -orbitgrowth Lemma3.7.1 Let A 1 A 2 A 3 beaLeonardtripleandlet ijk beapermutationof 123 .Thereare6validsequences 1 2 ;:::; ` suchthatweformachannelasin asinFigure3.5,namelythosewiththefollowing p i ;q i ; r i ofTheorem3.2.5where i { vi correspondtothelabeledsectionsofFigure3.6: i j,k;ii,k;ji,j;k... ii j,k;ii,j;ki,k;j... iii i,k;jj,k;ii,j;k... iv i,k;ji,j;kj,k;i... v i,j;kj,k;ii,k;j... vi i,j;ki,k;jj,k;i.... 39

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Figure3.6:6possiblechannelsfromaLeonardtriple Proof. SinceforeachLeonardtriplewehave3unorderedpairsandforeachofthese pairsthereisanantiautomorphismwhichxesbothelements,wecanstartwitha Leonardtripleandxthepair A i A j orthepair A i A k or A j A k .Continuingin this -orbitgrowthprocess,wehavetwopossiblepairswecanxineachofthethree newtriples.Thus,therearesixwayswecanstartastraightpathfromatriple.Once weareonastraightpath,toremainonitwemustrepeatthesameinitialsequenceof antiautomorphismsoflength3,otherwisethepathisnolongerstraight. 2 Denition3.7.2 Let A 1 A 2 A 3 beaLeonardtriple.Let 1 2 ;:::; ` beavalid sequenceofoneof6formsinLemma3.7.1.Wecalltheimagesof A 1 A 2 A 3 under thecompositionsofsuchvalidsequences channels .Observethatchannelscomein pairs. Denition3.7.3 Wesaythe lengthofachannel istheminimalpositivelength ` of asubsequenceofthevalidsequencedeningthechannelwhosecompositionxes A 1 A 2 A 3 .Wesaythethechannelis innite ifthereisnosuch ` Theorem3.7.4 Let A 1 A 2 A 3 beaLeonardtriple. 40

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i Ifthe -orbitisnite,theneverychannelofthecomplexhasnitelength. ii Ifsomechannelofthecomplexisinnite,thenthe -orbitisinnite. Proof. Supposethe -orbitisnite.ProceedtoformachannelasinDenition3.7.2. Sincethe -orbitisnite,eventuallysometriplemustberepeated.Thentheremust existsomepositivelength ` forsomevalidsubsequencesuchthat ` elementsofthe channelx A 1 A 2 A 3 .Sincewecandothisforeverychannel,itfollowsthatevery channelhasnitelengthasinDenition3.7.3andPartiholds.SincePartiiis thecontrapositiveofParti,theresultfollows. 2 Problem3.7.5 Proveordisprove:Assumethatthe -orbitofaLeonardtripleis vertex-niteandeverychannelhasnitelength.Thenthe -orbitisnite. Lemma3.7.6 Let A 1 A 2 A 3 beamodularLeonardtriple.Everychannelinthe -orbithasthesamelength. Proof. Since A 1 A 2 A 3 isamodularLeonardtriple,thereexistsanantiautomorphism whichxesoneelementofthetripleandswapstheremainingtwoelementsDenition 2.4.1.Thus,wehave S 3 actiononthetriple.Itfollowsthateachvertexmustappear inthesamenumberoftriples,henceeverynitechannelmusthavethesamenumber oftriples,andtheresultfollows. 2 Problem3.7.7 Proveordisprove:Assumethatthe -orbitofamodularLeonard tripleisvertex-niteandeverychannelhasnitelength.Thenthe -orbitisnite. Problem3.7.8 Proveordisprove:Let A 1 A 2 A 3 beaLeonardtriple.If jk i A i = A i ,then A 1 A 2 A 3 isaneequivalenttoamodularLeonardtriple. Lemma3.7.9 Ifavertex-nite -orbitofdegree m isnite,thenallchannelsof length m )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 canbewrittenasachanneloflengthatmost m )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 Proof. Assumethatthe -orbitofaLeonardtripleisvertex-nitewithdegree m .Sincethe -orbitisassumedtobenite,theneverychannelhasnitelength 41

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byTheorem3.7.4.ByTheorem4.2.1,byvertex-nitenessweknowthatvertices arerepeatedforspecialcompositionsofantiautomorphismsoflength m )]TJ/F19 11.9552 Tf 12.338 0 Td [(1.Thus, ifthe -orbitisnite,itmustbethecasethatallchannelsoflength m )]TJ/F19 11.9552 Tf 12.317 0 Td [(1canbe writtenasanotherchannelofshorterlength,otherwisewewouldhaveacontradiction. 2 Wedene Mathematica codeforperformingthechannelrotationsfromthissection. AlternatingSeq2[steps ; rst ; second ]:=Module[ f other=Select[ f 1 ; 2 ; 3 g ; And[# 6 =rst ; # 6 =second]&][[ 1 ]] g ; Table[If[Divisible[ i + 2 ; 3 ] ; rst ; If[Divisible[ i + 1 ; 3 ] ; second ; If[Divisible[ i ; 3 ] ; other]]] ; f i ; 1 ; steps g ]] ChannelRotate[steps ; rst ; second ]:= ApplyValidSequence[AlternatingSeq[steps ; rst ; second]] Herewegiveanexampleofachannelrotationoflength5. AlternatingSeq2[ 5 ; 1 ; 2 ] f 1 ; 2 ; 3 ; 1 ; 2 g 42

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3.8Problems Were-statetheopenproblemsfromthischapterinthissection. Problem3.5.11 Ifthe -orbitofaLeonardtripleisnite,mustthegraphofthe Leonardtriplebedistance-transitiveordistance-regular? Problem3.5.16 SupposethegraphofaLeonardtripleisregularandthatnoantiautomorphisminducesanyoftheautomorphismsofthegraph.Isthegrapharctransitive? Problem3.5.18 ObservethattheLeonardtriplesofan -orbitareeitherisomorphic oranti-isomorphic.IftwomodularLeonardtriplesisomorphic,aretheynecessarily inthesame -orbit? Problem3.6.7 AssumethateveryvertexinthecomplexofaLeonardtripleis vertex-nite.If2verticesofatriplehaveequaldegree,doesthethirdvertexhavethe samedegree? Problem3.7.5 Proveordisprove:Proveordisprove:Assumethatthe -orbitofa Leonardtripleisvertex-niteandeverychannelhasnitelength.Thenthe -orbit isnite. Problem3.7.7 Proveordisprove:Assumethatthe -orbitofamodularLeonard tripleisvertex-niteandeverychannelhasnitelength.Thenthe -orbitisnite. Problem3.7.8 Proveordisprove:Let A 1 A 2 A 3 beaLeonardtriple.If jk i A i = A i ,then A 1 A 2 A 3 isaneequivalenttoamodularLeonardtriple. 43

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4 -orbitsofmodularLeonardtriples Wenowstudythe -orbitsofmodularLeonardtriples.Weproceedconcretelyusing thedescriptionofeachtypefromLemmas2.6.3{2.6.9ofChapter2.Withthisdata wemaydescribethe -orbitsprecisely.WeshowthatmodularLeonardtriplesof typesII,IV,V,andVIhavejustonemodularLeonardtripleintheir -orbits.We showthatmodularLeonardtriplesoftypesO,I,andIIImayhaveniteorbitswhen theparameterstakeonspecialvalues,butformostvaluestheyareinnite.Sincethe computerwasusedtoverifycertainequalities,weincludesome Mathematica output asjusticationofourresults. 4.1TypesII,IV,V,andVI InthissectionweshowthatmodularLeonardtriplesoftypesII,IV,V,andVIhave trivial -orbits. Lemma4.1.1 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeII,IV,V,orVI. Then 12 3 = 13 2 = 23 1 .Thisantiautomorphismxeseach A 1 A 2 A 3 Proof. Wemaychooseabasiswithrespecttowhich A i isthetridiagonalmatrixofLemma2.6.2and jk i isrepresentedbythediagonalmatrix KNN with K N fromDenition2.7.1fortherespectivetypesII,IV,V,orVI.Thus, jk i A i = 44

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KNN )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A i t KNN = D )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 BD ,where D =diag ; b 0 2 1 c 1 ;:::; d )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 Y m =0 b m 2 m +1 c m +1 and B =tridiag 0 B B B @ c 1 :::c d )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 c d a 0 a 1 :::a d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 a d b 0 b 1 :::b d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 1 C C C A : Nowbymatrixmultiplication, h jk i A i i `;` +1 = D )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 `;` B `;` +1 D ` +1 ;` +1 = D )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 `;` A i ` +1 ;` D ` +1 ;` +1 = A i `;` +1 ; h jk i A i i `;` = D )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 `;` B `;` D `;` = A i t `;` = A i `;` ; h jk i A i i ` +1 ;` = D )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ` +1 ;` +1 B ` +1 ;` D `;` = D )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ` +1 ;` +1 A i `;` +1 D `;` = A i `;` = A i ` +1 ;` ; asshowninthefollowingcases. If A 1 A 2 A 3 isamodularLeonardtripleoftypeIIfromLemma2.6.5ofdiameter d .Thenwehavethefollowingcases. Case1: ` =1 h jk i A i i `;` +1 = )]TJ/F27 11.9552 Tf 9.299 0 Td [(hd s +2 d +4 4 = b 0 =[ A i ] `;` +1 ; h jk i A i i `;` = 0 + hd s +2 d +4 4 = a 0 =[ A i ] `;` ; h jk i A i i ` +1 ;` = h + s + d )]TJ/F27 11.9552 Tf 9.299 0 Td [(s )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 d 4+ s = c 1 =[ A i ] ` +1 ;` : Case2: 1 <`
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Case3: ` = d h jk i A i i `;` +1 = )]TJ/F28 7.9701 Tf 6.587 0 Td [(h d + s d +3 s +2 d +2 4 d + s )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = b d )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 =[ A i ] `;` +1 ; h jk i A i i `;` = 0 )]TJ/F30 7.9701 Tf 13.151 5.699 Td [()]TJ/F28 7.9701 Tf 6.587 0 Td [(h d + s d +3 s +2 d +2 4 d + s )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F30 7.9701 Tf 13.151 5.699 Td [()]TJ/F28 7.9701 Tf 6.586 0 Td [(h d )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 d + s + s 4 d + s )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = a d )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 =[ A i ] `;` ; h jk i A i i ` +1 ;` = )]TJ/F28 7.9701 Tf 10.494 5.699 Td [(hd s +2 4 = c d =[ A i ] ` +1 ;` : Case4: ` = d +1Theonlytermsthatappearsis h jk i A i i `;` = 0 + hd s +2 4 = a d =[ A i ] `;` : If A 1 A 2 A 3 isamodularLeonardtripleoftypeIVfromLemma2.6.7ofdiameter d .Then,for1 ` d h jk i A i i `;` +1 = 8 > < > : h ` )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 = 2 s + d +1if ` isodd ; h ` )]TJ/F28 7.9701 Tf 6.586 0 Td [(d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ` )]TJ/F28 7.9701 Tf 6.586 0 Td [(s +1 ` )]TJ/F28 7.9701 Tf 6.586 0 Td [(s= 2 if ` iseven ; = b ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 =[ A i ] `;` +1 ; h jk i A i i `;` = 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(c` )]TJ/F19 11.9552 Tf 11.955 0 Td [(1= a ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 =[ A i ] `;` ; h jk i A i i ` +1 ;` = 8 > < > : )]TJ/F28 7.9701 Tf 10.494 5.699 Td [(h` ` )]TJ/F28 7.9701 Tf 6.587 0 Td [(s + d +1 ` )]TJ/F28 7.9701 Tf 6.586 0 Td [(s= 2 if ` iseven ; )]TJ/F27 11.9552 Tf 9.298 0 Td [(h ` + s= 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1if ` isodd ; = c ` =[ A i ] ` +1 ;` : 46

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If A 1 A 2 A 3 isatypeVtripleasinLemma2.6.8ofdiameter d .Then,for 1 ` d h jk i A i i `;` +1 = 8 > < > : h ` )]TJ/F27 11.9552 Tf 11.955 0 Td [(d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1if ` isodd ; h ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(3 = 2 s + d +1 ` )]TJ/F28 7.9701 Tf 6.586 0 Td [(s ` )]TJ/F28 7.9701 Tf 6.587 0 Td [(s= 2 if ` iseven ; = b ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 =[ A i ] `;` +1 ; h jk i A i i `;` = 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(c ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = a ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 =[ A i ] `;` ; h jk i A i i ` +1 ;` = 8 > < > : )]TJ/F28 7.9701 Tf 10.494 5.698 Td [(h` ` + s= 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(d )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ` )]TJ/F28 7.9701 Tf 6.586 0 Td [(s= 2 if ` iseven ; )]TJ/F27 11.9552 Tf 9.298 0 Td [(h ` )]TJ/F27 11.9552 Tf 11.955 0 Td [(s + d +1if ` isodd ; = c ` =[ A i ] ` +1 ;` : If A 1 A 2 A 3 isatypeVItripleasinLemma2.6.9ofdiameter d .Then,for 1 ` d h jk i A i i `;` +1 = 8 > < > : h ` )]TJ/F28 7.9701 Tf 6.586 0 Td [(d )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(3 = 2 s + d +1 ` )]TJ/F28 7.9701 Tf 6.587 0 Td [(s= 2 if ` isodd ; h ` )]TJ/F27 11.9552 Tf 11.955 0 Td [(s if ` iseven ; = b ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 =[ A i ] `;` +1 ; h jk i A i i `;` = 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(c ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = a ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 =[ A i ] `;` ; h jk i A i i ` +1 ;` = 8 > < > : )]TJ/F27 11.9552 Tf 9.299 0 Td [(h` if ` iseven ; )]TJ/F28 7.9701 Tf 10.494 5.699 Td [(h` ` )]TJ/F28 7.9701 Tf 6.587 0 Td [(s + d +1 ` + s= 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(d )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ` )]TJ/F28 7.9701 Tf 6.586 0 Td [(s= 2 if ` isodd ; = c ` =[ A i ] ` +1 ;` : Thisallowsustoconcludethat 12 3 A 3 = A 3 13 2 A 2 = A 2 23 1 A 1 = A 1 regardlessofthediameterchosen.Since jk i xes A j and A k bydenition,itfollows thateach jk i xesallthreeelements A 1 A 2 A 3 .Hence,theymustbethesame antiautomorphism.Thus 12 3 = 13 2 = 23 1 2 Onecanalsoverifytheseresultsin Mathematica .Hereweincludetheoutputfor thetypeIIcase,butomittypesIV,V,andVIsincetheyaresimilar. Type= 2 ; d = 2 ; 47

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ZeroMatrixQ[Simplify[Factor[alpha[ 1 ; A [ 1 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 1 ]]]] True ZeroMatrixQ[Simplify[Factor[alpha[ 2 ; A [ 2 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 2 ]]]] True ZeroMatrixQ[Simplify[Factor[alpha[ 3 ; A [ 3 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 3 ]]]] True Corollary4.1.2 The -orbitofanymodularLeonardtriplesoftypesII,IV,V,and VIconsistsofjustonemodularLeonardtriple. Proof. FollowsdirectlyfromLemma4.1.1. 2 Lemma4.1.3 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeOorIII.Then 12 3 = 13 2 = 23 1 andthisantiautomorphismxeseach A 1 A 2 A 3 onlywhen = 1 Proof. WeproceedasintheproofofLemma4.1.1andcomputethefollowingmatrix entries.Bymatrixmultiplication,if A 1 A 2 A 3 isamodularLeonardtripleoftype OfromLemma2.6.3ofdiameter d =1,then h jk i A i i `;` +1 = D )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 `;` A i ` +1 ;` D ` +1 ;` +1 = 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( + 2 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 2 = 2 b ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ; h jk i A i i `;` = A i t `;` = A i `;` ; h jk i A i i ` +1 ;` = D )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ` +1 ;` +1 A i `;` +1 D `;` = 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 0 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = c ` 2 : Inthesameway,if A 1 A 2 A 3 isamodularLeonardtripleoftypeIIIfromLemma 2.6.6ofdiameter d ,then h jk i A i i `;` +1 = D )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 `;` A i ` +1 ;` D ` +1 ;` +1 = h 2 ` )]TJ/F27 11.9552 Tf 11.955 0 Td [(d )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( + 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 = 2 b ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ; h jk i A i i `;` = A i t `;` = A i `;` ; h jk i A i i ` +1 ;` = D )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ` +1 ;` +1 A i `;` +1 D `;` = h` 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2 = c ` 2 : 48

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Thus,fortypeOandtypeIIItriples,iftheantiautomorphismisthesameandxes allthreeelements,thenthefollowingequalitiesmusthold: h jk i A i i `;` +1 = A i `;` +1 2 b ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = b ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ; h jk i A i i `;` = A i `;` ; h jk i A i i ` +1 ;` = A i ` +1 ;` c ` 2 = c ` : Hence,thishappensonlywhen 2 =1 = 1andtheresultfollows. 2 Lemma4.1.4 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeI.Then 12 3 13 2 and 23 1 areeachdistinctanddonotxthesameelementsoftheoriginaltriple. Proof. WeproceedasintheproofofLemma4.1.1andcomputethefollowingmatrix entries.Bymatrixmultiplication,if A 1 A 2 A 3 isamodularLeonardtripleoftypeI fromLemma2.6.4ofdiameter d ,thenfor1 ` d h jk i A i i `;` +1 = )]TJ/F27 11.9552 Tf 10.494 8.087 Td [( 2 q 2 h )]TJ/F27 11.9552 Tf 11.955 0 Td [(q d )]TJ/F28 7.9701 Tf 6.587 0 Td [(` +1 q )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 q ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 + 3 q d + ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 q d )]TJ/F28 7.9701 Tf 6.586 0 Td [(` +1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(q ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 q 2 ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(3 = 2 q 2 b ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ; h jk i A i i `;` = A i t `;` = A i `;` ; h jk i A i i ` +1 ;` = h )]TJ/F27 11.9552 Tf 11.955 0 Td [(q ` + q d )]TJ/F28 7.9701 Tf 6.587 0 Td [(` )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 q d + ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 2 q d )]TJ/F28 7.9701 Tf 6.586 0 Td [(` +3 )]TJ/F27 11.9552 Tf 11.955 0 Td [(q ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 q 2 ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = c ` 2 q 2 : Thus,fortypeItriples,iftheantiautomorphismisthesameandxesallthree elements,thenthefollowingequalitiesmusthold: h jk i A i i `;` +1 = A i `;` +1 2 q 2 b ` )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = b ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ; h jk i A i i `;` = A i `;` ; h jk i A i i ` +1 ;` = A i ` +1 ;` c ` 2 q 2 = c ` : Howeverthisrequiresthat 2 q 2 =1whichcannothappenaccordingtoLemma2.6.4. Thus,theresultholds. 2 49

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Lemma4.1.5 Let A 1 A 2 A 3 beamodularLeonardtriple.Thefollowingareequivalent: i jk i = ik j forsomepermutation ijk of 123 ii 12 3 = 13 2 = 23 1 iii Either A 1 A 2 A 3 isoftypeOorIIIwhere = 1 or A 1 A 2 A 3 isoftypeII, IV,V,orVI. iv jk i xes A i forsomepermutationof 1 ; 2 ; 3 Proof. ByLemmas4.1.1,4.1.3,and4.1.4,wehavethatPartiiiimpliesPartii. Clearly,PartiiimpliesParti.Bythedenitionofeach jk i inTheorem2.7.3and byspecialvaluesof K N ,and R foreachtypeofmodularLeonardtripleinLemmas 2.7.4{2.7.9,itfollowsthatPartiimpliesPartiii.Since jk i xes A i meansthat jk i xesallthreeelementsofthetriple,PartivimpliesPartii.Ontheotherhand,if 12 3 = 13 2 = 23 1 holds,theneachantiautomorphism jk i mustxallthreeelements including A i .HencePartiiimpliesPartivandwehavetherequiredequivalence. 2 WenowreneLemma3.1.5. Corollary4.1.6 WithreferencetoLemma4.1.5, i IftheequivalentconditionsofLemma4.1.5hold,thenthereisonly1valid sequenceoflength1andnolongervalidsequence. ii IftheequivalentconditionsofLemma4.1.5donothold,thenthereareexactly 3 2 ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 validsequencesoflength ` ` 1 Proof. PartifollowsdirectlyfromLemma4.1.5.PartiifollowsfromLemma 3.1.5since 12 3 6 = 13 2 6 = 23 1 2 Theremainderofourdiscussionfocusesonthemoreinteresting -orbitsarising frommodularLeonardtriplesoftypesO,I,andIII. 50

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4.2TypesOandIII ModularLeonardtriplesoftypesOandIIIdependprimarilyonasingleparameter .Weshowthatwhen isaprimitive3 rd ,4 th ,or5 th rootofunity,the -orbits formthetetrahedron,octahedron,andicosahedron,respectively.Finally,wegive anexampleofaninnite -orbitwhen isaprimitive7 th rootofunity.Webegin bycharacterizingthevertex-nitenessconditionforthesetypesofmodularLeonard triples.Wesplittheoddandevencases. Theorem4.2.1 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeOortypeIII. Let beasinLemma2.6.3orLemma2.6.6respectively.Let 1 2 ;:::; m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 be avalidsequencesuchthat h A i = A i .Assume f h A j ; h A k g6 = f A j ;A k g for 1 h m )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 .Thenthefollowinghold: i Suppose m 3 isodd.Then m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 A j = A k and m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A k = A j ifandonlyif isaprimitive m th rootofunity. ii Suppose m 2 iseven.Then m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A j = A j and m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A k = A k ifandonly if isaprimitive m th rootofunity. Proof. Let A 1 A 2 A 3 beatypeOortypeIIImodularLeonardtriple.Supposethat for m 3odd, m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 A j = A k and m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 A k = A j andthatfor m 2even, m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A j = A j and m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A k = A k .Then,when m 3odd,itfollowsthatthecorresponding entriesof m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A j and A k mustbeequalandthecorrespondingentriesof m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 A k and A j mustbeequal.Similarly, m 2even,correspondingentriesof m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 A j and A j mustbeequal.Thenweformthefollowingsystemofequationsregardlessofthe diameterchosen: P m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 i =0 i )]TJ/F27 11.9552 Tf 11.955 0 Td [( + 2 =0 ; P m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 i =0 i =0 : Bytherestrictionfor foundinLemmas2.6.3and2.6.6respectively,weknowthat 1 )]TJ/F27 11.9552 Tf 12.322 0 Td [( + 2 6 =0.Thus,thesystemisreducedtothesingleinequality: P m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 i =0 =0. Thus,1+ + 2 + ::: + m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 =0,hence isaprimitive m th rootofunity. 51

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Suppose isaprimitive m th rootofunity.Let 1 2 ::: ` beavalidsequence ofatypeIIItriple A 1 A 2 A 3 .ByLemma3.1.5,thereare3 2 ` )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 validsequencesof antiautomorphisms 1 2 ::: ` andbyLemma3.6.1,only2suchsequencesbring youbacktotheoriginaltriple.Itfollowsthat ` )]TJ/F19 11.9552 Tf 9.929 0 Td [(1= m )]TJ/F19 11.9552 Tf 9.929 0 Td [(2 ` = m )]TJ/F19 11.9552 Tf 9.93 0 Td [(1.ByDenition 3.1.2,thismeansthatafter m )]TJ/F19 11.9552 Tf 11.465 0 Td [(1stagesofgrowth,weproducethesameelementsas thepreviousstages.If m 3odd,wegetthat m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 A j = A k and m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 A k = A j If m 2even, m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 A k = A k and m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 A j = A j SincetypeOtriplesareaspecial caseoftypeIIItriples,thesameproofholdsforthesetriples. 2 Corollary4.2.2 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeOortypeIII. Let beasinLemma2.6.3orLemma2.6.6respectively.The -orbitofthetripleis vertex-niteifandonlyif isaprimitiverootofunity. Proof. ClearfromTheorem4.2.1. 2 ThenextsetofexamplesillustrateTheorem4.2.1forsmallprimitiverootsof unity.Moreover,thesearespecialsincethetriplesintheseexampleshavenite -orbitswhichformgeometricsolids. Example4.2.3 Considerthecasewhen = 1,thatislet beaprimitive2 nd root ofunity.Thenwehavethefollowingequalitieswhichformatrianglecomplex: A 1 = 23 1 A 1 ;A 2 = 13 2 A 2 ;A 3 = 12 3 A 3 : Figure4.1:TypeO/IIIcomplex 2 =1 52

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TheidentitiesinExample4.2.3areveriedin Mathematica asfollows.Wespecify thetypeanddiameter d andlet beaprimitive2 nd rootofunity.Notethatweomit typeIIIcodesinceitfollowssimilarlyastypeO. Type= 0 ; d = 1 ; nu= )]TJ/F54 11.9552 Tf 9.298 0 Td [(1 ; ZeroMatrixQ[Simplify[Factor[alpha[ 1 ; A [ 1 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 1 ]]]] True ZeroMatrixQ[Simplify[Factor[alpha[ 2 ; A [ 2 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 2 ]]]] True ZeroMatrixQ[Simplify[Factor[alpha[ 3 ; A [ 3 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 3 ]]]] True Example4.2.4 Considerthecasewhen = )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 i p 3 2 ,thatislet beaprimitive 3 rd rootofunity.Let A 4 = 23 1 A 1 ,thenbyTheorem4.2.1,wehavethefollowing equalities: A 1 = 23 1 A 4 ; A 2 = 13 2 A 4 ; A 3 = 12 3 A 4 ; A 4 = 13 2 A 2 = 12 3 A 3 : Thustheelementsofthe -orbitareniteandformatetrahedronFigure4.2a withverticesandfaceslistedinTable4.2bwhen 3 =1. TheidentitiesinExample4.2.4areveriedin Mathematica asfollows.Wespecify thetypeanddiameter d let beaprimitive3 rd rootofunity.Notethatweomittype IIIcodesinceitfollowssimilarlyastypeO. 53

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aTetrahedron Faces A 1 A 2 A 3 A 1 A 3 A 4 A 1 A 2 A 4 A 2 A 3 A 4 b Figure4.2:TypeO/IIIcomplex 3 =1 type= 0 ; d = 1 ; nu= )]TJ/F55 7.9701 Tf 6.587 0 Td [(1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(Sqrt[ )]TJ/F55 7.9701 Tf 6.587 0 Td [(1 ]Sqrt[ 3 ] 2 ; Wedene A 4 intermsof A 1 A [ 4 ]=alpha[ 1 ; A [ 1 ]]; ZeroMatrixQ[Simplify[Factor[alpha[ 1 ; A [ 4 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 1 ]]]] True ZeroMatrixQ[Simplify[Factor[alpha[ 2 ; A [ 4 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 2 ]]]] True ZeroMatrixQ[Simplify[Factor[alpha[ 3 ; A [ 4 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 3 ]]]] True 54

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Example4.2.5 Considerthecasewhen = i ,thatislet beaprimitive4 th root ofunity.Let A 4 = 23 1 A 1 A 5 = 13 2 A 2 A 6 = 12 3 A 3 .ThenbyTheorem4.2.1, wehavethefollowingequalities: A 1 = 23 1 A 4 ;A 4 = 13 2 A 4 = 12 3 A 4 ; A 2 = 13 2 A 5 ;A 5 = 23 1 A 5 = 12 3 A 5 ; A 3 = 12 3 A 6 ;A 6 = 23 1 A 6 = 13 2 A 6 : Thustheelementsofthe -orbitareniteandformanoctahedronFigure4.3a withverticesandfaceslistedinTable4.3bwhen 4 =1. aOctahedron Faces A 1 A 2 A 3 A 1 A 2 A 6 A 1 A 3 A 5 A 1 A 5 A 6 A 2 A 3 A 4 A 2 A 4 A 6 A 3 A 4 A 5 A 4 A 5 A 6 b Figure4.3:TypeO/IIIcomplex 4 =1 55

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TheidentitiesinExample4.2.5areveriedin Mathematica asfollows.Wespecify thetypeanddiameter d let beaprimitive4 th rootofunity.Notethatweomittype IIIcodesinceitfollowssimilarlyastypeO. type= 0 ; d = 1 ; nu= )]TJ/F19 11.9552 Tf 9.298 0 Td [(Sqrt[ )]TJ/F54 11.9552 Tf 9.299 0 Td [(1 ]; Wedene A 4 { A 6 intermsofprevious A i 's. A [ 4 ]=alpha[ 1 ; A [ 1 ]]; A [ 5 ]=alpha[ 2 ; A [ 2 ]]; A [ 6 ]=alpha[ 3 ; A [ 3 ]]; ZeroMatrixQ[Simplify[Factor[alpha[ 1 ; A [ 4 ]] )]TJ/F54 11.9552 Tf 11.955 0 Td [(A [ 1 ]]]] True ZeroMatrixQ[Simplify[Factor[alpha[ 2 ; A [ 5 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 2 ]]]] True ZeroMatrixQ[Simplify[Factor[alpha[ 3 ; A [ 6 ]] )]TJ/F54 11.9552 Tf 11.956 0 Td [(A [ 3 ]]]] True Example4.2.6 Considerthecasewhen 2 p 5 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 4 i q 5+ p 5 8 ; )]TJ 6.586 6.598 Td [(p 5 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 4 i q 5 )]TJ 6.586 6.598 Td [(p 5 8 thatislet beaprimitive5 th rootofunity.Let A 4 = 23 1 A 1 A 5 = 13 2 A 2 A 6 = 12 3 A 3 A 7 = 23 1 A 6 A 8 = 13 2 A 4 A 9 = 12 3 A 5 A 10 = 23 1 A 9 A 11 = 13 2 A 7 A 12 = 12 3 A 8 .ThenbyTheorem4.2.1,wehavethefollowingequalities: A 1 = 23 1 A 4 ;A 7 = 12 3 A 4 ; A 2 = 13 2 A 5 ;A 8 = 23 1 A 5 ; A 3 = 12 3 A 6 ;A 9 = 13 2 A 6 ; A 4 = 13 2 A 8 = 12 3 A 7 ;A 10 = 13 2 A 10 = 12 3 A 10 ; A 5 = 23 1 A 8 = 12 3 A 9 ;A 11 = 23 1 A 11 = 12 3 A 11 ; A 6 = 23 1 A 7 = 13 2 A 9 ;A 12 = 23 1 A 12 = 13 2 A 12 : Thusthe -orbitisniteanditselementsformanicosahedronFigure4.4awith verticesandfaceslistedinTable4.4bwhen 5 =1. 56

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aIcosahedron Faces A 1 A 2 A 3 A 1 A 2 A 6 A 1 A 3 A 5 A 1 A 5 A 9 A 1 A 6 A 9 A 2 A 3 A 4 A 2 A 4 A 7 A 2 A 6 A 7 A 3 A 4 A 8 A 3 A 5 A 8 A 4 A 7 A 10 A 4 A 8 A 10 A 5 A 8 A 11 A 5 A 9 A 11 A 6 A 7 A 12 A 6 A 9 A 12 A 7 A 10 A 12 A 8 A 10 A 11 A 9 A 11 A 12 A 10 A 11 A 12 b Figure4.4:TypeO/IIIcomplex 5 =1 57

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TheidentitiesinExample4.2.6areveriedin Mathematica asfollows.Wespecify thetypeanddiameter d let beaprimitive5 th rootofunity.Notethatweomittype IIIcodesinceitfollowssimilarlyastypeO. type= 0 ; d = 1 ; nu= Sqrt[ 5 ] )]TJ/F55 7.9701 Tf 6.586 0 Td [(1 4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(Sqrt h 5 +Sqrt[ 5 ] 8 i Sqrt[ )]TJ/F54 11.9552 Tf 9.298 0 Td [(1 ]; Wedene A 4 { A 12 intermsofprevious A i 's. A [ 4 ]=alpha[ 1 ; A [ 1 ]]; A [ 5 ]=alpha[ 2 ; A [ 2 ]]; A [ 6 ]=alpha[ 3 ; A [ 3 ]]; A [ 7 ]=alpha[ 1 ; A [ 6 ]]; A [ 8 ]=alpha[ 2 ; A [ 4 ]]; A [ 9 ]=alpha[ 3 ; A [ 5 ]]; A [ 10 ]=alpha[ 1 ; A [ 9 ]]; A [ 11 ]=alpha[ 2 ; A [ 7 ]]; A [ 12 ]=alpha[ 3 ; A [ 8 ]]; ZeroMatrixQ[Simplify[Factor[ A [ 1 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 1 ; A [ 4 ]]]]] True ZeroMatrixQ[Simplify[Factor[ A [ 2 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 2 ; A [ 5 ]]]]] True ZeroMatrixQ[Simplify[Factor[ A [ 3 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 3 ; A [ 6 ]]]]] True Lemma4.2.7 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeOortypeIII.Then cannotbeaprimitive 6 th rootofunity. Proof. ByLemmas2.6.3and2.6.6,we 2 )]TJ/F27 11.9552 Tf 12.037 0 Td [( +1 6 =0.Thus,byquadraticformula, 6 = 1 i p 3 2 .Thus, cannotbeaprimitive6 th rootofunity. 2 Considerthefollowingexampleofaninnite -orbitdespitehavingvertex-niteness. Example4.2.8 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeOortypeIII. Let beaprimitive7 th rootofunity.Thenwehaveavertex-nite -orbitasin Denition3.6.2.ByLemma3.7.9,inorderforthe -orbittobenite,itmustbethe casethatallchannelsoflength m )]TJ/F19 11.9552 Tf 11.469 0 Td [(1=6canbewrittenaschainofashorterlength duetovertex-niteness.However,thisisnotthecasesinceitcanbeveriedthat 23 1 13 2 12 3 13 1 13 2 12 3 A 3 doesnotequalanyotherelementofthe -orbitoflength lessthan6.Thus,weconjecturethatthe -orbitof A 1 A 2 A 3 isinnite.Our 58

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computationsgivethe7-regulartessellationofthehyperbolicplaneforthecomplex ofthistripleSeeFigure4.5,uptoeightstagesofgrowthandweexpectthisto continue. Figure4.5:Conjecturedinnitecomplex 7 =1 TheresultsinExample4.2.8areveriedin Mathematica asfollows.Wespecify thetypeanddiameter d let beaprimitive7 th rootofunity.Weagainomittype IIIcodesinceitfollowssimilarlyastypeO. type= 0 ; d = 1 ;nu= )]TJ/F19 11.9552 Tf 9.299 0 Td [( )]TJ/F54 11.9552 Tf 9.298 0 Td [(1 5 = 7 ; x =alpha[ 1 ; alpha[ 2 ; alpha[ 3 ; alpha[ 1 ; alpha[ 2 ; alpha[ 3 ; A[ 3 ]]]]]]]; Simplify[ x )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 1 ; alpha[ 2 ; alpha[ 1 ; alpha[ 2 ; alpha[ 1 ; alpha[ 2 ; A[ 2 ]]]]]]]] False Simplify[ x )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 1 ; alpha[ 2 ; alpha[ 1 ; alpha[ 2 ; alpha[ 1 ; alpha[ 3 ; A[ 3 ]]]]]]]] False Simplify[ x )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 1 ; alpha[ 2 ; alpha[ 1 ; alpha[ 2 ; alpha[ 3 ; alpha[ 1 ; A[ 1 ]]]]]]]] False Problem4.2.9 Showthatthe -orbitofExample4.2.8isinnite. Problem4.2.10 ShowthatthecomplexofExample4.2.8isthe7-regulartessellation ofthehyperbolicplane. 59

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Problem4.2.11 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeOortypeIII. Let beasinLemma2.6.3orLemma2.6.6respectively.Proveordisprove:For m> 6,if isaprimitive m th rootofunity,thenthe -orbitisinnite. Observethatregularityisanecessaryconditiononthestructureofthecomplexes ofmodularLeonardtriples.Forobjectsofgenus0,theonlycomplexesofmodularLeonardtriplesarethetriangleandtheplatonicsolids.Therearevarioussolids withtriangularfaceswhicharenotregularsuchasthetriangulardipyramidFigure 4.6a,thepentagonaldipyramidFigure4.6b,thegyroelongatedsquaredipyramidFigure4.6c,thetriaugmentedtriangularprismFigure4.6d,andthesnub disphenoidFigure4.6e.ThesecomplexesareverymuchlikethoseofaLeonard triple,butcannotbeoneofthemodularLeonardtriples. a J 12 b J 13 c J 17 d J 51 e J 84 Figure4.6:Non-regularsolidswithtriangularfaces Problem4.2.12 DoanycomplexesofLeonardtriplesformoneofthenon-regular JohnsonsolidsFigure4.6? 60

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4.3TypeI The -orbitoftypeItriplesisneverjustatriangleasshowninLemma4.1.5.Modular LeonardtriplesoftypesIdependprimarilyonparameters and q .However, and q arealsodependentuponthediameter d ofthetriple.Wewillshowthatthe -orbitof typeItriplesmaybeniteforspecialvaluesof and q .Weconsidervertex-niteness. Lemma4.3.1 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeIwithdiameter d Then and q cannotbethesameprimitive m th rootsofunityifeitherofthefollowing hold: i m 2f 2 ; 3 ; 4 ; 5 ;:::; 2 d g ;or ii m 2f 2 d +4 ; 2 d +6 ; 2 d +8 ;:::; 4 d +2 g Proof. Weprovethecontrapositive.Let and q beasinLemma2.6.4.Sincethe diameterofmodularLeonardtriplesofTypeImustbeatleast2,wedonotconsider thecasewhere m< 2.ByLemma2.6.4,wehave 2 q i 6 =1,for0 i 2 d )]TJ/F19 11.9552 Tf 12.049 0 Td [(2.Thus, if = q ,then 2+ i 6 =1.Hence, cannotbeaprimitive+ i th rootofunity.Then m 6 =2+ i ,thatis m= 2f 2 ; 3 ; 4 ; 5 ;:::; 2 d g .TheotherrestrictionfromLemma2.6.4 wasthat 3 q d + i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 6 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1.If = q ,thenwehave d + i +2 6 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1.Squaringbothsidesof thisinequalityyields 2 d +2 i +4 6 =1.Thus, cannotbeaprimitive d +2 i +4 th root ofunity.Then m 6 = d +2 i +4,thatis m= 2f 2 d +4 ; 2 d +6 ; 2 d +8 ;:::; 4 d +2 g Hencetheresultholds. 2 Conjecture4.3.2 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeI.The -orbit isvertex-niteifandonlyif = q isaprimitive m th rootofunity. Wegivesomeexamplesofnite -orbitsoftypeItriples. Example4.3.3 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeIofdiameter d =2.Let and q beasinLemma2.6.4.ByLemma4.3.1,wecanlet = q 2 p 5 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 4 i q 5+ p 5 8 ; )]TJ 6.587 6.598 Td [(p 5 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 4 i q 5 )]TJ 6.587 6.598 Td [(p 5 8 ,thatislet and q bethesameprimitive5 th rootofunity.ThenwehavethesameequalitiesasinExample4.2.6.Hence,wethe -orbitisniteandthecomplexformedistheicosahedronshowninFigure4.4a. 61

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TheidentitiesinExample4.3.3areveriedin Mathematica asfollows.Wespecify thetypeanddiameter d let beaprimitive5 th rootofunity. type= 1 ; d = 2 ; nu= Sqrt[ 5 ] )]TJ/F55 7.9701 Tf 6.586 0 Td [(1 4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(Sqrt h 5 +Sqrt[ 5 ] 8 i Sqrt[ )]TJ/F54 11.9552 Tf 9.298 0 Td [(1 ]; Wedene A 4 { A 12 intermsofprevious A i 's.. A [ 4 ]=alpha[ 1 ; A [ 1 ]]; A [ 5 ]=alpha[ 2 ; A [ 2 ]]; A [ 6 ]=alpha[ 3 ; A [ 3 ]]; A [ 7 ]=alpha[ 1 ; A [ 6 ]]; A [ 8 ]=alpha[ 2 ; A [ 4 ]]; A [ 9 ]=alpha[ 3 ; A [ 5 ]]; A [ 10 ]=alpha[ 1 ; A [ 9 ]]; A [ 11 ]=alpha[ 2 ; A [ 7 ]]; A [ 12 ]=alpha[ 3 ; A [ 8 ]]; ZeroMatrixQ[Simplify[Factor[ A [ 1 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 1 ; A [ 4 ]]]]] True ZeroMatrixQ[Simplify[Factor[ A [ 2 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 2 ; A [ 5 ]]]]] True ZeroMatrixQ[Simplify[Factor[ A [ 3 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 3 ; A [ 6 ]]]]] True Example4.3.4 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeIofdiameter d =2.ByLemma4.3.1,wecanlet = q = 1 i p 3 2 ,thatisaprimitive6 th rootof unity.Let A 4 = 23 1 A 1 A 5 = 13 2 A 2 A 6 = 12 3 A 3 A 7 = 23 1 A 6 A 8 = 23 1 A 5 A 9 = 13 2 A 4 .Thenwehaveanite -orbitfromthefollowingequalities: A 1 = 23 1 A 4 ;A 6 = 23 1 A 7 = 13 2 A 7 ; A 2 = 13 2 A 5 ;A 7 = 23 1 A 6 = 13 2 A 6 = 12 3 A 7 ; A 3 = 12 3 A 6 ;A 8 = 23 1 A 5 = 12 3 A 5 = 13 2 A 8 ; A 4 = 13 2 A 9 = 12 3 A 9 ;A 9 = 13 2 A 4 = 12 3 A 4 = 23 1 A 9 : A 5 = 23 1 A 8 = 12 3 A 8 ; Thecomplexformedisatriangletoruswithvertices1{9ineachcrosssectionFigure 4.7acorrespondingtothesubscriptsofthe A 0 i s 62

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aCross-sectionoftorus Faces A 1 A 2 A 3 A 1 A 2 A 6 A 1 A 3 A 5 A 1 A 5 A 8 A 1 A 6 A 7 A 1 A 7 A 8 A 2 A 3 A 4 A 2 A 4 A 8 A 2 A 6 A 9 b Facescont'd A 2 A 8 A 9 A 3 A 4 A 7 A 3 A 5 A 9 A 3 A 7 A 9 A 4 A 5 A 6 A 4 A 5 A 8 A 4 A 6 A 7 A 5 A 6 A 9 A 7 A 8 A 9 c Figure4.7:TypeIcomplex 6 = q 6 =1 TheidentitiesinExample4.3.4areveriedin Mathematica asfollows. type= 1 ; d = 2 ;nu= q = 1 +Sqrt[ )]TJ/F55 7.9701 Tf 6.587 0 Td [(3 ] 2 ; Wedene A 4 { A 9 intermsofprevious A i 's. A [ 4 ]=alpha[ 1 ; A [ 1 ]]; A [ 5 ]=alpha[ 2 ; A [ 2 ]]; A [ 6 ]=alpha[ 3 ; A [ 3 ]]; A [ 7 ]=alpha[ 1 ; A [ 6 ]]; A [ 8 ]=alpha[ 1 ; A [ 5 ]]; A [ 9 ]=alpha[ 2 ; A [ 4 ]]; ZeroMatrixQ[Simplify[Factor[ A [ 1 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 1 ; A [ 4 ]]]]] True ZeroMatrixQ[Simplify[Factor[ A [ 2 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 2 ; A [ 5 ]]]]] True ZeroMatrixQ[Simplify[Factor[ A [ 3 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 3 ; A [ 6 ]]]]] True Example4.3.5 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeIofdiameter d =2or3.Let and q beasinLemma2.6.4.ByLemma4.3.1,wecanlet = q =be aprimitive7 th rootofunity.Let A 4 = 23 1 A 1 A 5 = 13 2 A 2 A 6 = 12 3 A 3 A 7 = 23 1 A 5 A 8 = 23 1 A 6 A 9 = 13 2 A 4 A 10 = 13 2 A 6 A 11 = 12 3 A 4 A 12 = 12 3 A 5 63

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A 13 = 23 1 A 9 A 14 = 23 1 A 10 A 15 = 23 1 A 11 A 16 = 23 1 A 12 A 17 = 13 2 A 8 A 18 = 13 2 A 12 A 19 = 23 1 A 17 A 20 = 23 1 A 18 A 21 = 13 2 A 14 A 22 = 13 2 A 15 A 23 = 12 3 A 13 ,and A 24 = 12 3 A 16 .Thenwehaveanite -orbitfromthefollowing equalities: A 1 = 23 1 A 4 ;A 13 = 13 2 A 7 = 12 3 A 23 ; A 2 = 13 2 A 5 ;A 14 = 12 3 A 9 = 13 2 A 21 ; A 3 = 12 3 A 6 ;A 15 = 12 3 A 8 = 13 2 A 22 ; A 4 = 13 2 A 9 = 12 3 A 11 ;A 16 = 13 2 A 11 = 12 3 A 24 ; A 5 = 23 1 A 7 = 12 3 A 12 ;A 17 = 12 3 A 7 = 23 1 A 19 ; A 6 = 23 1 A 8 = 13 2 A 10 ;A 18 = 12 3 A 10 = 23 1 A 20 ; A 7 = 13 2 A 13 = 12 3 A 17 ;A 19 = 13 2 A 19 = 12 3 A 19 ; A 8 = 13 2 A 17 = 12 3 A 15 ;A 20 = 13 2 A 20 = 12 3 A 20 ; A 9 = 23 1 A 13 = 12 3 A 14 ;A 21 = 23 1 A 21 = 12 3 A 21 ; A 10 = 23 1 A 14 = 12 3 A 18 ;A 22 = 23 1 A 22 = 12 3 A 22 ; A 11 = 23 1 A 15 = 13 2 A 16 ;A 23 = 23 1 A 23 = 13 2 A 23 ; A 12 = 23 1 A 16 = 13 2 A 18 ;A 24 = 23 1 A 24 = 13 2 A 24 : Thecomplexformedisatriangletoruswithvertices1{24ineachcrosssectionFigure 4.8acorrespondingtothesubscriptsofthe A 0 i s 64

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aCross-sectionoftorus Facesofcomplex A 1 A 2 A 3 A 1 A 2 A 6 A 1 A 3 A 5 A 1 A 5 A 10 A 1 A 10 A 18 A 2 A 12 A 18 A 2 A 6 A 12 A 2 A 3 A 4 A 2 A 4 A 8 A 2 A 8 A 15 A 2 A 11 A 15 A 2 A 6 A 11 A 3 A 4 A 7 A 3 A 7 A 13 A 3 A 9 A 13 A 3 A 5 A 9 A 4 A 7 A 14 A 4 A 14 A 20 A 4 A 16 A 20 A 4 A 8 A 16 A 5 A 9 A 17 A 5 A 17 A 22 A 5 A 16 A 22 A 5 A 10 A 16 A 6 A 11 A 17 A 6 A 12 A 14 A 6 A 14 A 23 A 6 A 17 A 23 b Facescont'd A 7 A 12 A 22 A 7 A 13 A 19 A 7 A 14 A 22 A 7 A 19 A 22 A 8 A 10 A 16 A 8 A 10 A 23 A 8 A 15 A 19 A 8 A 19 A 23 A 9 A 11 A 17 A 9 A 11 A 20 A 9 A 13 A 21 A 9 A 20 A 21 A 10 A 18 A 21 A 10 A 21 A 23 A 11 A 15 A 24 A 11 A 20 A 24 A 12 A 18 A 24 A 12 A 22 A 24 A 13 A 15 A 18 A 13 A 15 A 19 A 13 A 18 A 21 A 14 A 20 A 21 A 14 A 21 A 23 A 15 A 18 A 24 A 16 A 20 A 24 A 16 A 22 A 24 A 17 A 19 A 22 A 17 A 19 A 23 c Figure4.8:TypeIcomplex 7 = q 7 =1 65

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TheidentitiesinExample4.3.5areveriedin Mathematica asfollows.Wespecify thetypeanddiameter d let beaprimitive7 th rootofunity. type= 1 ; d = 2 ; nu= )]TJ/F19 11.9552 Tf 9.298 0 Td [( )]TJ/F54 11.9552 Tf 9.298 0 Td [(1 5 = 7 ; q = )]TJ/F19 11.9552 Tf 9.299 0 Td [( )]TJ/F54 11.9552 Tf 9.299 0 Td [(1 5 = 7 ; Wedene A 4 { A 2 4intermsofprevious A i 's. A [ 4 ]=alpha[ 1 ; A [ 1 ]]; A [ 5 ]=alpha[ 2 ; A [ 2 ]]; A [ 6 ]=alpha[ 3 ; A [ 3 ]]; A [ 7 ]=alpha[ 1 ; A [ 5 ]]; A [ 8 ]=alpha[ 1 ; A [ 6 ]]; A [ 9 ]=alpha[ 2 ; A [ 4 ]]; A [ 10 ]=alpha[ 2 ; A [ 6 ]]; A [ 11 ]=alpha[ 3 ; A [ 4 ]]; A [ 12 ]=alpha[ 3 ; A [ 5 ]]; A [ 13 ]=alpha[ 1 ; A [ 9 ]]; A [ 14 ]=alpha[ 1 ; A [ 10 ]]; A [ 15 ]=alpha[ 1 ; A [ 11 ]]; A [ 16 ]=alpha[ 1 ; A [ 12 ]]; A [ 17 ]=alpha[ 2 ; A [ 8 ]]; A [ 18 ]=alpha[ 2 ; A [ 12 ]]; A [ 19 ]=alpha[ 1 ; A [ 17 ]]; A [ 20 ]=alpha[ 1 ; A [ 18 ]]; A [ 21 ]=alpha[ 2 ; A [ 14 ]]; A [ 22 ]=alpha[ 2 ; A [ 15 ]]; A [ 23 ]=alpha[ 3 ; A [ 13 ]]; A [ 24 ]=alpha[ 3 ; A [ 16 ]]; ZeroMatrixQ[Simplify[Factor[ A [ 1 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 1 ; A [ 4 ]]]]] True ZeroMatrixQ[Simplify[Factor[ A [ 2 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 2 ; A [ 5 ]]]]] True ZeroMatrixQ[Simplify[Factor[ A [ 3 ] )]TJ/F19 11.9552 Tf 11.955 0 Td [(alpha[ 3 ; A [ 6 ]]]]] True Problem4.3.6 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeIofanydiameter d 2.Let and q beasinLemma2.6.4orLemma2.6.4.Proveordisprove:For m 6,if = q isaprimitive m th rootofunity,thenthe -orbitisniteandformsa torus. 66

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4.4Problems Were-statetheopenproblemsfromthischapterinthissection. Problem4.2.9 Showthatthe -orbitofExample4.2.8isinnite. Problem4.2.10 ShowthatthecomplexofExample4.2.8isthe7-regulartessellationofthehyperbolicplane. Problem4.2.11 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeOortype III.Let beasinLemma2.6.3orLemma2.6.6respectively.Proveordisprove:For m> 6,if isaprimitive m th rootofunity,thenthe -orbitisinnite. Problem4.2.12 DoanycomplexesofLeonardtriplesformoneofthenon-regular JohnsonsolidsFigure4.6? Problem4.3.2 Provetheconjectureofthisnumber. Problem4.3.6 Let A 1 A 2 A 3 beamodularLeonardtripleoftypeIofanydiameter d 2.Let and q beasinLemma2.6.4orLemma2.6.4.Proveordisprove: For m 6,if = q isaprimitive m th rootofunity,thenthe -orbitisniteand formsatorus. 67

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5AssociationschemesfrommodularLeonardtriples WenowturnourattentiontoanotherproblemconcerningmodularLeonardtriples. Todiscussthisproblemwewillneedsomefurtherbackgroundmaterialconcerning Kroneckerproducts,associationschemesandrelatednotions. 5.1Preliminariesforassociationschemes WerecallthenotionofaKroneckerproduct. Denition5.1.1 [23]If A isan m -byn matrixand B isa p -byq matrix,thenthe Kroneckerproduct A B isthe mp -bynq blockmatrix A B = 2 6 6 6 4 a 11 B:::a 1 n B . . . . a m 1 B:::a mn B 3 7 7 7 5 : .1.1 Lemma5.1.2 If A B C ,and D arematricesofsuchsizethatonecanformthe matrixproducts AC and BD ,then A B C D existsand A B C D = AC BD: .1.2 Proof. ExpandtheKroneckerproductsasshownin.1.1andapplyregularmatrix multiplication. 2 Lemma5.1.3 [3]Supposethat A and B aresquarematricesofsize n and q respectively.Let 1 ,..., n betheeigenvaluesof A and 1 ,..., q bethoseof B listedwith 68

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algebraicmultiplicity.Thenthetraceandspectrumof A B arerespectively tr A B =tr A tr B ; .1.3 spec A B = f i j j i =1 ;:::;n;j =1 ;:::;q g : .1.4 Denition5.1.4 [3]Let X beaset. i The Cartesian squareof X istheset X X = f ; j 2 X; 2 X g ii Let R X X .Its dualsubset istheset R 0 = f ; j ; 2 X g iii The diagonal subsetisthesetDiag X = f !;! : 2 X g Werecallthenotionofanassociationscheme. Denition5.1.5 [3]Bya k-classsymmetricassociationscheme wemeanasetof points, X ,alongwith k +1binaryrelations R 0 R 1 ,..., R k ,called associateclasses whichpartition X X suchthat i R 0 =Diag X ; ii R i = R 0 i ;i =1 ;:::;k ; iii forall i;j;` in f 0 ;:::;k g ,thereisaninteger p ` ij suchthat,forall ; 2 R ` jf 2 X : ; 2 R i ; ; 2 R j gj = p ` ij : Denition5.1.6 [3]WithreferencetoDenition5.1.5,wecallthenumberofassociateclasses,inthiscase k +1,the rank oftheassociationscheme.Elements ; 2 X arecalled i-thassociates if ; 2 R i .Theparameters p ` ij arecalledthe intersection numbers ofanassociationscheme.Notethatthesuperscript ` doesnotsignifya power. WedenethenotionofaHammingassociationscheme. Denition5.1.7 [3]Let)-343(bean n -setandlet=)]TJ/F28 7.9701 Tf 87.332 4.338 Td [(m .For and in,let and be i -thassociatesif and dierinexactly i positions,where0 i m .These classeson formthe Hammingscheme ,H m n ,onthe m -thpowerofan n set,becausethe Hammingdistance between and isdenedtobethenumberof positionsinwhichtheydier. 69

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Figure5.1:The3-dimensionalbinaryHammingcube H ; 2 WegiveanexampleofaHammingscheme. Example5.1.8 Considertheset X of3-tuplesofthe2-set f 0 ; 1 g .Wethinkofthe HammingschemeH,2asagraphwithvertexsetXandanedgebetweentwo verticesifandonlyiftheydierbyexactlyoneentryFigure5.1,[25].Theshortest pathbetweenverticeswillthenindicatewhichassociateclasstheyarecontainedin. 5.2ThePaulimatrices InsomesensemodularLeonardtriplesgeneralizethePaulimatrices,althoughthere areothergeneralizations[37].Inthissectionweexaminehownearthemodular LeonardtriplesaretotheGrassmaniancodesofA.Roy[39],[40]intermsofa relatedconstructionofassociationschemes.ThePaulimatricesarealsoexamplesof Roy'swork. Example5.2.1 The Paulimatrices arethefollowing2 2matrices: 1 = 0 @ 01 10 1 A ; 2 = 0 @ 0 )]TJ/F27 11.9552 Tf 9.298 0 Td [(i i 0 1 A ; 3 = 0 @ 10 0 )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 1 A ; where i = p )]TJ/F19 11.9552 Tf 9.298 0 Td [(1.Inquantummechanics, 1 and 2 oftendenotepositionandmomentumvectors,while 3 issimplytheproductof 1 and 2 multipliedbythescalar 1 i [37]. TheyareHermitianandunitary[40],suchthatfor i =1,2,3, i tr i =0; 70

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ii det i =-1; iii 2 i = 0 @ 10 01 1 A = I: Theset f 1 ; 2 ; 3 ;I g formsanorthogonalbasisfortherealHilbertspaceof2 2 complexHermitianmatrices[37]. Lemma5.2.2 WithreferencetoExample5.2.1, 1 2 3 formamodularLeonard tripleoftypeO. Proof. Notethat 1 isoftheform.6.4,where b 0 =1, c 1 =1, a 0 = a 1 =0, 2 isof theform.6.6,where 1 = )]TJ/F27 11.9552 Tf 9.298 0 Td [(i ,and 3 isoftheform.6.5,where 0 =1, 1 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1. Then,when = 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [( + 2 0 )]TJ/F28 7.9701 Tf 6.586 0 Td [( 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 2 = 2 i )]TJ/F28 7.9701 Tf 6.586 0 Td [(i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 2 = 2 i 2 i =1= b 0 ; 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [( 0 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 2 = 2 i 2 i =1= c 1 ; 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b 0 =1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1=0= a 0 ; 0 )]TJ/F27 11.9552 Tf 11.955 0 Td [(c 1 =1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1=0= a 1 : ThusbyLemma2.6.3, 1 2 3 formamodularLeonardtripleoftypeO. 2 ObservethattypesI{VIexcludedthecasewhen d =1inorderforeverymodular Leonardtripletohaveauniquedescription.Onecantake d =1ineachtypeexcept typeVandVIandendupwithamemberoftypeO. 71

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5.3APartitionofKroneckerproducts Inthissectionweconsidercertainassociationschemeswhoseconstructioninvolves Kroneckerproducts. Denition5.3.1 LetMat n K bethesetof n n matriceswith n 2 Z + .Givena niteset A ofmatricesfromMat n K andapositiveinteger k ,dene X k A = f A 1 A 2 A k j A i 2Ag : .3.5 Denition5.3.2 Givenaniteset A ofmatricesfromMat n K andapositiveinteger k ,dene ST k A = f tr B 1 B 2 j B 1 ;B 2 2 X k A g ; SS k A = f spec B 1 B 2 j B 1 ;B 2 2 X k A g : Denition5.3.3 aDeneapartition T k A = f t j t 2 ST k A g of X k A X k A bytherelation B 1 t B 2 iftr B 1 B 2 = t: bDeneapartition k A = f s j s 2 SS k A g of X k A X k A bytherelation B 1 s B 2 ifspec B 1 B 2 = s: Notethatthematricesinthe -orbitofamodularLeonardtriplearesimilarand hencecospectral.Weshallmeetthefollowingspecialsituations. Lemma5.3.4 Let A beanitesetofcospectralmatricesandlet T 0 ;T 1 betherespectiveconstantssuchthat tr A i A j = 8 > < > : T 0 ifi 6 = j; T 1 ifi=j : .3.6 72

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Suppose B 1 B 2 2 X k A aregivenby B 1 = A 1 A 2 A k ; B 2 = A 0 1 A 0 2 A 0 k : Let u = jf l j A l = A 0 l gj v = jf l j A l 6 = A 0 l gj .Then tr B 1 B 2 = T u 0 T v 1 : Proof. By.1.2, B 1 B 2 = A 1 A 0 1 A 2 A 0 2 A k A 0 k .Thusby.1.3,tr B 1 B 2 =tr A 1 A 0 1 tr A 2 A 0 2 tr A k A 0 k andtheresultfollowsfrom.3.6. 2 Lemma5.3.5 Let A beanitesetofcospectralmatricesandlet S 0 ;S 1 betherespectiveconstantssuchthat spec A i A j = 8 > < > : S 0 if i 6 = j; S 1 ifi=j : .3.7 Suppose B 1 B 2 2 X k A aregivenby B 1 = A 1 A 2 A k ; B 2 = A 0 1 A 0 2 A 0 k : Let u = jf l j A l = A 0 l gj v = jf l j A l 6 = A 0 l gj .Then spec B 1 B 2 = S u 0 S v 1 : Proof. By.1.2, B 1 B 2 = A 1 A 0 1 A 2 A 0 2 A k A 0 k .Thusby.1.4,spec B 1 B 2 =spec A 1 A 0 1 spec A 2 A 0 2 spec A k A 0 k andtheresultfollowsfrom.3.7. 2 73

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5.4HammingSchemesfrommodularLeonardtriples Webeginthissectionbystatingexplicitythepossibletracesandspectraforthe productsofcanonicalmodularLeonardtriples. Lemma5.4.1 Let A 1 A 2 A 3 beacanonicalmodularLeonardtriple.Sincethese matricesarecospectral,thereexistconstantsinthesenseofLemmas5.3.4and5.3.5 suchthat T 0 = f z 0 0 + z 1 1 + + z d d j z i 2 K g ; .4.8 T 1 = 2 0 + 2 1 + + 2 d ; .4.9 S 0 = f )]TJ/F27 11.9552 Tf 9.299 0 Td [(b i )]TJ/F27 11.9552 Tf 11.955 0 Td [(c i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + b i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + c i +1 + 0 i j 0 i d g ; .4.10 S 1 = f i j j 0 i;j d g ; .4.11 where i a i b i ,and c i arethematrixelementsfromLemma2.6.2. Proof. Since A i and A j aresquarematrices,itfollowsthattr A i A j =tr A j A i Bythemodularityofthesematrices,equations.4.8and.4.10hold.Toshow thatspec A i A j = S 0 when i 6 = j ,wemustcomputethesematrixproductsforeach ofthesixfamiliesofcanonicalmodularLeonardtriples[13].Startingwithbasecase ofdiameterd=5andcontinuingbyinduction,result.4.10holds.Fromtheshape ofthesematrices,itfollowsthatspec A 2 1 =spec A 2 2 =spec A 2 3 = f 2 0 ; 2 1 ; ; 2 d g = S 1 andresult.4.11holds. 2 Thenexttheoremshowsthatthetracesandspectraofproductsofmodular Leonardtriplespartitiontheset X k A in.3.5,formingspecialassociationschemes. Theorem5.4.2 Let A = f A 1 ;A 2 ;A 3 g beamodularLeonardtriple.Then T k A and k A areassociationschemesforallk 1.Inparticular, T k A and k A are isomorphictotheHammingschemeH k ,3. Proof. ByDenition5.3.3, T k A and k A partition X k A X k A wherethe relations B 1 t B 2 iftr B 1 B 2 = t and B 1 s B 2 ifspec B 1 B 2 = s arenecessarilysymmetricsincetr B 1 B 2 =tr B 2 B 1 andspec B 1 B 2 =spec B 2 B 1 andthereexists 74

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appropriateintersectionnumberswhichsatisfytheremainingconditionsforassociationschemesinDenition5.1.5.Moreover,sincefor and in T k A and are i -thassociatesif and dierinexactly i positions,where0 i k ,weseethat T k A isinfactisomorphictoaHammingscheme[3].Sincespec B 1 B 2 = S u 0 S v 1 B 1 and B 2 areinthesame i -threlationiftheydierinexactly i places.Thus, k A is alsoisomorphictoaHammingscheme.Finally,since X k A iscomposedofelements oflength k chosenfromasetofsize3,theresultfollows. 2 Problem5.4.3 DoesTheorem5.4.2holdif A = f A 1 ;A 2 ;A 3 g isonlyaLeonardtriple withoutmodularity? Problem5.4.4 DoesTheorem5.4.2holdforaneshiftsofmodularLeonardtriples? Corollary5.4.5 Let A 1 A 2 A 3 beamodularLeonardtriple.Let ijk beapermutationof123.Let A = A i ; ik j A j ;A k .Then, T k A and k A areassociation schemesforallk 1.Inparticular, T k A and k A areisomorphictotheHamming schemeH k ,3. Proof. FollowsdirectlyfromLemma2.3.2andTheorem5.4.2. 2 IntheworkofRoy,weseethathisGrassmaniancodesprovideexampleswhere T k A and k A arenotisomorphic. 75

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5.5Counterexamplesfornon-modulartriples WhilewehaveshownthatthesetofKroneckerproductsof k elementsofamodular Leonardtriple A 1 A 2 A 3 formsassociationschemesusingthespectrumandtrace, thisresultdoesnotextendtoalltriplesconsistingoftwotridiagonalandonediagonal matrices.ThisisbecauseoftheimportantmodularitypropertyofmodularLeonard triples.Considerthefollowingcounterexamples. Example5.5.1 Let A 1 A 2 A 3 beamodularLeonardtriple.Considerthetriple A 1 A 2 A T 3 ,where A T 3 isthetransposematrixof A 3 .Then,whilespec A 2 1 =spec A 2 2 =spec A T 3 A T 3 = f 2 0 ; 2 1 ; ; 2 d g = S 1 andwhiletr A 2 1 =tr A 2 2 =tr A T 3 A T 3 = f 2 0 + 2 1 + + 2 d g = T 1 duetotheshapeofthematrices,wehavespec A 1 A T 3 6 =spec A T 3 A 2 =spec A 1 A 2 = S 0 andtr A 1 A T 3 6 =tr A T 3 A 2 =tr A 1 A 2 = T 0 .Thus, thesetofKroneckerproductsofkelementsoftriple A 1 A 2 A T 3 ,donotforma Hammingschemedespitesimilarshapesofthematrices.Thisisbecausethetripleis notmodular.Byasimilarargument,thetriple A T 1 A 2 A 3 failsaswell.Infact,they donotgiverisetoanyassociationschemeingeneral. Example5.5.2 Let A 1 A 2 A 3 beamodularLeonardtriple.Considerthetriple A 1 A 2 A T 3 ,where A T 3 istheconjugatetransposematrixof A 3 .Liketheprevious example,spec A 1 A T 3 6 =spec A T 3 A 2 =spec A 1 A 2 = S 0 andtr A 1 A T 3 6 =tr A T 3 A 2 =tr A 1 A 2 = T 0 .Thus,thesetofKroneckerproductsofkelementsoftriple A 1 A 2 A T 3 doesnotformanassociationscheme. 76

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5.6Problems Problem5.4.3 DoesTheorem5.4.2holdif A = f A 1 ;A 2 ;A 3 g isonlyaLeonardtriple? Problem5.4.4 DoesTheorem5.4.2holdforaneshiftsofmodularLeonardtriples? Problem5.6.1 Consideranite -orbitofamodularLeonardtriple.Althoughnot anassociationscheme,wehavesomeregularstructure.Whatisit? Problem5.6.2 Consideranitesubsetofan -orbitofamodularLeonardtriple. Althoughnotanassociationscheme,wehavesomeregularstructure.Whatisit? Problem5.6.3 Considerthe k th stateofgrowthofanite -orbitofamodular Leonardtriple.Althoughnotanassociationscheme,wehavesomeregularstructure. Whatisit? 77

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References [1]R.AskeyandJ.Wilson,Asetoforthogonalpolynomialsthatgeneralizethe Racahcoecientsor6-jsymbols. SIAMJ.Math.Anal. 10 ,10081016. [2]R.AskeyandJ.Wilson,Somebasichypergeometricorthogonalpolynomialsthat generalizetheJacobipolynomials. Mem.Amer.Math.Soc. 54 [3]R.A.Bailey,AssociationSchemes,DesignedExperiments",AlgebraandCombinatorics,CambridgeUniversityPress,Cambridge,2004. [4]E.BannaiandT.Ito,AlgebraicCombinatoricsI",Benjamin/Cummings,Menlo Park,1984. [5]A.E.Brouwer,A.M.Cohen,andA.Neumaier,Distance-RegularGraphs. Springer,NewYork,1989. [6]P.Baseilhac,Anintegrablestructurerelatedwithtridiagonalalgebras, Nuclear Phys.B 705 ,no.3,605-619. [7]P.BaseilhacandK.Koizumi,AdeformedanalogueofOnsager'ssymmetryin the XXZ openspinchain, J.Stat.Mech.TheoryExp. 10 ,1-15. [8]P.BaseilhacandK.Koizumi,Anewinnite-dimensionalalgebraforquantum integrablemodels, NuclearPhys.B 720 ,no.3,325-347. [9]J.S.CaughmanandN.Wol,TheTerwilligeralgebraofadistance-regulargraph thatsupportsaspinmodel, J.AlgebraicCombin. 21 ,no.3,289-310. [10]B.Curtin,Distance-regulargraphswhichsupportaspinmodelarethin. Discr. Math. 197 198 ,205-216. 78

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[24]D.Leonard,Orthogonalpolynomials,duality,andassociationschemes. SIAMJ. Math.Anal. 13 ,656663. [25]S.Mckinley,TheHammingCodesandDelsarte'sLinearProgrammingBound, availableat http://www.mth.pdx.edu/ ~ caughman/thesis.pdf [26]S.`Miklavic,Leonardtriplesandhypercubes,preprint,availableat http: //arxiv.org/abs/0705.0518 [27]K.NomuraandP.Terwilliger,AnetransformationsofaLeonardpair,preprint ,availableat http://arxiv.org/abs/math/0611783 [28]K.Nomura,TridiagonalpairsandtheAskey-Wilsonrelations, LinearAlgebra Appl. 397 ,99-106. [29]K.Nomura,Arenementofthesplitdecompositionofatridiagonalpair, Linear AlgebraAppl. 403 ,1-23. [30]K.Nomura,Tridiagonalpairsofheightone, LinearAlgebraAppl. 403 118-142. [31]K.NomuraandP.Terwilliger,Sometraceformulaeinvolvingthesplitsequences ofaLeonardpair. LinearAlgebraAppl. 413 ,189-201. [32]K.NomuraandP.Terwilliger,Thedeterminantof AA )]TJ/F27 11.9552 Tf 10.703 0 Td [(A A foraLeonardpair A;A LinearAlgebraAppl. 416 ,880-889. [33]K.NomuraandP.Terwilliger,Matrixunitsassociatedwiththesplitbasisofa Leonardpair, LinearAlgebraAppl. 418 ,775-787. [34]K.NomuraandP.Terwilliger,BalancedLeonardpairs, LinearAlgebraAppl. 420 ,no.1,51-69. [35]K.NomuraandP.Terwilliger,Lineartransformationsthataretridiagonalwith respecttobotheigenbasesofaLeonardpair, LinearAlgebraAppl. 420 no.1,198-207. [36]K.NomuraandP.Terwilliger,TheswitchingelementforaLeonardpair, Linear AlgebraAppl. 428 ,1083-1108. 80

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AbouttheAuthor AftertakingmyrstcollegemathclassatEdisonStateCollege,Iwasinspiredtoteach mathematics.MysubsequenteducationalcareeratFloridaGulfCoastUniversity andtheUniversityofSouthFloridahasaordedmenumerousopportunitestogrow asbothastudentandteacherofmathematics.Writingthisthesishasgivenme anewfoundappreciationofthediscipline.Ilookforwardtopassingalongtomy studentsatleastaglimpseofthebeautyandusefulnessofmathematics.


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Some combinatorial structures constructed from modular Leonard triples
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ABSTRACT: Let V denote a vector space of finite positive dimension. An ordered triple of linear operators on V is said to be a Leonard triple whenever for each choice of element of the triple there exists a basis of V with respect to which the matrix representing the chosen element is diagonal and the matrices representing the other two elements are irreducible tridiagonal. A Leonard triple is said to be modular whenever for each choice of element there exists an antiautomorphism of End(V) which fixes the chosen element and swaps the other two elements. We study combinatorial structures associated with Leonard triples and modular Leonard triples. In the first part we construct a simplicial complex of Leonard triples. The simplicial complex of a Leonard triple is the smallest set of linear operators which contains the given Leonard triple with the property that if two elements of the set are part of a Leonard triple, then the third element of the triple is also in the set. In the second part we construct a Hamming association scheme from modular Leonard triples using a method used previously in the context of Grassmanian codes.
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Leonard pairs
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Hamming association scheme
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