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Association Schemes1 Association Schemes1 Chris Godsil Combinatorics & Optimization University of Waterloo ©2010 1June 3, 2010 ii Preface These notes provide an introduction to association schemes, along with some related algebra. Their form and content has benefited from discussions with Bill Martin and Ada Chan. iii iv Contents Preface iii 1 Schemes and Algebras1 1.1 Definitions and Examples........................1 1.2 Strongly Regular Graphs.........................3 1.3 The Bose-Mesner Algebra........................6 1.4 Idempotents................................7 1.5 Idempotents for Association Schemes.................9 2 Parameters 13 2.1 Eigenvalues................................ 13 2.2 Strongly Regular Graphs......................... 15 2.3 Intersection Numbers.......................... 16 2.4 Krein Parameters............................. 17 2.5 The Frame Quotient........................... 20 3 An Inner Product 23 3.1 An Inner Product............................. 23 3.2 Orthogonal Projection.......................... 24 3.3 Linear Programming........................... 25 3.4 Cliques and Cocliques.......................... 28 3.5 Feasible Automorphisms........................ 30 4 Products and Tensors 33 4.1 Kronecker Products............................ 33 4.2 Tensor Products.............................. 34 4.3 Tensor Powers............................... 37 4.4 Generalized Hamming Schemes.................... 38 4.5 A Tensor Identity............................. 39 v vi CONTENTS 4.6 Applications................................ 41 5 Subschemes and Partitions 43 5.1 Equitable Partitions............................ 43 5.2 Subschemes and Partitions....................... 45 5.3 Primitivity................................. 48 5.4 Simple Subsets.............................. 50 5.5 Completely Regular Subsets....................... 51 6 Translation Schemes 53 6.1 Characters................................. 53 6.2 Translation Graphs............................ 55 6.3 Translation Schemes and their Duals................. 56 6.4 Linear Graphs............................... 58 6.5 Geometry, Codes and Graphs...................... 59 6.6 Language.................................. 61 7 Duality 63 7.1 The Discrete Fourier Transform..................... 63 7.2 The Hadamard Transform........................ 65 7.3 Two Matrix Duals............................. 68 7.4 MacWilliams Theorem.......................... 69 7.5 Projective Planes............................. 71 7.6 Duality................................... 73 7.7 Duality and Type II Matrices...................... 75 7.8 Difference Sets.............................. 76 8 Type-II Matrices 79 8.1 Type-II Matrices.............................. 79 8.2 Two Algebras................................ 81 8.3 Eigenspaces................................ 83 9 Galois Theory 85 9.1 Bose-Mesner Automorphisms..................... 85 9.2 Galois.................................... 86 9.3 Applications................................ 89 9.4 Multipliers................................. 91 CONTENTS vii 10 A Bestiary 99 10.1 Cyclic Schemes.............................. 99 10.2 Paley Graphs................................ 100 10.3 Quasisymmetric Designs........................ 102 10.4 Partial Spreads............................... 104 10.5 Covers of Complete Bipartite Graphs................. 106 10.6 Groups................................... 108 11 Algebra and Modules 111 11.1 Algebras.................................. 111 11.2 Division Algebras............................. 112 11.3 Maps and Modules............................ 115 11.4 Opposites................................. 116 11.5 Schur’s Lemma.............................. 117 12 Semisimple Modules 119 12.1 Summands and Idempotents...................... 119 12.2 Primary Decomposition......................... 121 12.3 Group Algebras.............................. 122 12.4 Semisimple Modules........................... 124 12.5 Semisimple Modules: Examples.................... 125 12.6 Indecomposable Modules........................ 127 13 Semisimple Algebras 131 13.1 Semisimple Algebras........................... 131 13.2 Simple Artinian Algebras......................... 133 13.3 Composition Series............................ 135 13.4 Semisimple Artinian Algebras...................... 137 13.5 Representations.............................. 138 13.6 Centralizers................................ 139 13.7 Trace.................................... 141 13.8 Maschke.................................. 141 14 Division Algebras 145 14.1 Central Simple Algebras......................... 145 14.2 Factors................................... 148 14.3 Finite Division Algebras......................... 149 14.4 Real Algebra................................ 151 viii CONTENTS 15 Work 153 15.1 Classical Parameters........................... 153 16 Adjacency Algebras 155 16.1 Extending the Adjacency Algebra.................... 155 16.2 Some Applications............................ 156 16.3 Cospectral Awful Graphs......................... 157 16.4 Modules and Walks............................ 160 16.5 An Inner Product on Polynomials................... 161 16.6 Spectral Decomposition......................... 162 16.7 Orthogonal Polynomials......................... 163 16.8 Distance-Regular Graphs........................ 166 16.9 Locally Distance-Regular Graphs.................... 166 16.10Coherent Algebras............................ 168 17 Line Digraphs 171 17.1 Line Digraphs............................... 171 17.2 Quantum Walks.............................. 173 17.3 Eigenvalues of Quantum Walks..................... 173 18 Lie Algebras 177 18.1 Basics.................................... 177 18.2 Enveloping Algebras........................... 179 18.3 Posets.................................... 179 18.4 Representations of Lie Algebras..................... 181 18.5 Bilinear Forms............................... 182 18.6 An Example................................ 184 18.7 Irreducible Modules........................... 185 18.8 Semisimple Elements.......................... 188 18.9 Semisimple Modules........................... 190 19 Terwilliger Algebras 193 19.1 Modules.................................. 193 19.2 Thinness.................................. 195 19.3 Jaeger Algebras.............................. 196 20 Strongly Regular Graphs 199 20.1 Strongly Regular Graphs......................... 199 20.2 Local Eigenvalues............................. 202 CONTENTS ix 20.3 Dimensions................................ 204 21 Hamming Schemes 207 21.1 The Binary Hamming Scheme..................... 207 21.2 Modules.................................. 208 22 Spin 211 22.1 Braids.................................... 211 22.2 Nomura Algebras............................. 211 22.3 Braids.................................... 212 22.4 Jones Pairs................................. 213 22.5 Gauge Equivalence............................ 216 22.6 Nomura Algebras of Type-II matrices................. 216 22.7 Spin Models................................ 218 23 Abelian Spin 221 23.1 Schemes.................................. 221 23.2 Coordinate Matrices........................... 222 23.3 Duality................................... 224 23.4 Modular Invariance............................ 226 23.5 The Cyclic Group............................. 227 x CONTENTS Chapter 1 Schemes and Algebras Our first three chapters provide an introduction to the basic theory of associ- ation schemes and to some of their applications. In this chapter we introduce association schemes and describe their structure. 1.1 Definitions and Examples We try to motivate the definitions to come. Suppose X is a graph with vertex set V and diameter d. For i 1,...,d we define X to be the graph with vertex set V , Æ i where two vertices are adjacent in X if they are at distance i in X . (So X X .) i Æ 1 Let Ai denote the adjacency matrix of Xi , set A0 equal to I and consider the matrix algebra C[A ] over C generated by A1,..., Ad . If we identify the automorphism group of X with the set of permutation ma- trices that commute with A1, then each automorphism of X lies in the commu- tant of C[A ]. Thus, for example, if C[A ] Matn n(C), then the automorphism Æ £ group of X must be the identity group. Since the matrices A0,..., Ad are lin- early independent, C[A ] has dimension at least d 1. This suggests that the Å case where dim(C[A ]) d 1 should be interesting. In fact the dimension of Æ Å C[A ] is d 1 if and only if the matrices A ,..., A form an association scheme. Å 0 d An association scheme with d classes is a set A {A ,..., A } of 01-matrices Æ 0 d such that (a) A I. 0 Æ Pd (b) i 0 Ai J. Æ Æ 1 2 CHAPTER 1. SCHEMES AND ALGEBRAS (c) AT A for each i. i 2 (d) A A A A span(A ). i j Æ j i 2 Note that (b) implies that the matrices A0,..., Ad are linearly independent, and (d) that the algebra they generate has dimension d 1. Since J is the sum of the Å Ai , it commutes with each Ai , which implies that all rows and columns of Ai have the same sum. An association scheme is symmetric if each matrix in it is symmetric. We view A1,..., Ad as adjacency matrices of directed graphs X1,..., Xd , with com- mon vertex set V . We say two vertices u and v are i-related if uv is an arc in Xi . 1.1.1 Example. The Johnson scheme J(v,k). The vertex set of this scheme is the set of all k-subsets of a fixed set of v elements. Two vertices ® and ¯ are i-related if ® ¯ k i. This scheme has k classes. j \ j Æ ¡ 1.1.2 Example. The Grassman scheme Jq (v,k). The vertex set is the set of all
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