Table Of Content

1 Association Schemes ChrisGodsil Combinatorics&Optimization UniversityofWaterloo ©2010 1June 3, 2010 ii Preface Thesenotesprovideanintroductiontoassociationschemes,alongwithsome related algebra. Their form and content has beneﬁted from discussions with BillMartinandAdaChan. iii iv Contents Preface iii 1 SchemesandAlgebras 1 1.1 DeﬁnitionsandExamples . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 StronglyRegularGraphs . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 TheBose-MesnerAlgebra . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 IdempotentsforAssociationSchemes . . . . . . . . . . . . . . . . . 9 2 Parameters 13 2.1 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 StronglyRegularGraphs . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 IntersectionNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 KreinParameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 TheFrameQuotient . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 AnInnerProduct 23 3.1 AnInnerProduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 OrthogonalProjection . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 LinearProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 CliquesandCocliques . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5 FeasibleAutomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 30 4 ProductsandTensors 33 4.1 KroneckerProducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 TensorProducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 TensorPowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.4 GeneralizedHammingSchemes . . . . . . . . . . . . . . . . . . . . 38 4.5 ATensorIdentity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 v vi CONTENTS 4.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 SubschemesandPartitions 43 5.1 EquitablePartitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 SubschemesandPartitions . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Primitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.4 SimpleSubsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.5 CompletelyRegularSubsets . . . . . . . . . . . . . . . . . . . . . . . 51 6 TranslationSchemes 53 6.1 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.2 TranslationGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.3 TranslationSchemesandtheirDuals . . . . . . . . . . . . . . . . . 56 6.4 LinearGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.5 Geometry,CodesandGraphs . . . . . . . . . . . . . . . . . . . . . . 59 6.6 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7 Duality 63 7.1 TheDiscreteFourierTransform. . . . . . . . . . . . . . . . . . . . . 63 7.2 TheHadamardTransform . . . . . . . . . . . . . . . . . . . . . . . . 65 7.3 TwoMatrixDuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.4 MacWilliamsTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.5 ProjectivePlanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.7 DualityandTypeIIMatrices . . . . . . . . . . . . . . . . . . . . . . 75 7.8 DifferenceSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 8 Type-IIMatrices 79 8.1 Type-IIMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.2 TwoAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.3 Eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 9 GaloisTheory 85 9.1 Bose-MesnerAutomorphisms . . . . . . . . . . . . . . . . . . . . . 85 9.2 Galois . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 9.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 9.4 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 CONTENTS vii 10 ABestiary 99 10.1 CyclicSchemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 10.2 PaleyGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 10.3 QuasisymmetricDesigns . . . . . . . . . . . . . . . . . . . . . . . . 102 10.4 PartialSpreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 10.5 CoversofCompleteBipartiteGraphs . . . . . . . . . . . . . . . . . 106 10.6 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 11 AlgebraandModules 111 11.1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 11.2 DivisionAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 11.3 MapsandModules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 11.4 Opposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 11.5 Schur’sLemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 12 SemisimpleModules 119 12.1 SummandsandIdempotents . . . . . . . . . . . . . . . . . . . . . . 119 12.2 PrimaryDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . 121 12.3 GroupAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 12.4 SemisimpleModules . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 12.5 SemisimpleModules: Examples . . . . . . . . . . . . . . . . . . . . 125 12.6 IndecomposableModules . . . . . . . . . . . . . . . . . . . . . . . . 127 13 SemisimpleAlgebras 131 13.1 SemisimpleAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 13.2 SimpleArtinianAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . 133 13.3 CompositionSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 13.4 SemisimpleArtinianAlgebras . . . . . . . . . . . . . . . . . . . . . . 137 13.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 13.6 Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 13.7 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 13.8 Maschke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 14 DivisionAlgebras 145 14.1 CentralSimpleAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . 145 14.2 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 14.3 FiniteDivisionAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . 149 14.4 RealAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 viii CONTENTS 15 Work 153 15.1 ClassicalParameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 16 AdjacencyAlgebras 155 16.1 ExtendingtheAdjacencyAlgebra . . . . . . . . . . . . . . . . . . . . 155 16.2 SomeApplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 16.3 CospectralAwfulGraphs . . . . . . . . . . . . . . . . . . . . . . . . . 157 16.4 ModulesandWalks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 16.5 AnInnerProductonPolynomials . . . . . . . . . . . . . . . . . . . 161 16.6 SpectralDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . 162 16.7 OrthogonalPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . 163 16.8 Distance-RegularGraphs . . . . . . . . . . . . . . . . . . . . . . . . 166 16.9 LocallyDistance-RegularGraphs . . . . . . . . . . . . . . . . . . . . 166 16.10CoherentAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 17 LineDigraphs 171 17.1 LineDigraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 17.2 QuantumWalks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 17.3 EigenvaluesofQuantumWalks . . . . . . . . . . . . . . . . . . . . . 173 18 LieAlgebras 177 18.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 18.2 EnvelopingAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 18.3 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 18.4 RepresentationsofLieAlgebras . . . . . . . . . . . . . . . . . . . . . 181 18.5 BilinearForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 18.6 AnExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 18.7 IrreducibleModules . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 18.8 SemisimpleElements . . . . . . . . . . . . . . . . . . . . . . . . . . 188 18.9 SemisimpleModules . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 19 TerwilligerAlgebras 193 19.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 19.2 Thinness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 19.3 JaegerAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 20 StronglyRegularGraphs 199 20.1 StronglyRegularGraphs . . . . . . . . . . . . . . . . . . . . . . . . . 199 20.2 LocalEigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 CONTENTS ix 20.3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 21 HammingSchemes 207 21.1 TheBinaryHammingScheme . . . . . . . . . . . . . . . . . . . . . 207 21.2 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 22 Spin 211 22.1 Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 22.2 NomuraAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 22.3 Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 22.4 JonesPairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 22.5 GaugeEquivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 22.6 NomuraAlgebrasofType-IImatrices . . . . . . . . . . . . . . . . . 216 22.7 SpinModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 23 AbelianSpin 221 23.1 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 23.2 CoordinateMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 23.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 23.4 ModularInvariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 23.5 TheCyclicGroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 x CONTENTS

Association Schemes [Lecture notes] PDF

242 Pages·2010·1.048 MB·English

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