Algebraic Combinatorics I: Association Schemes

Algebraic Combinatorics I......Page 1
Preface......Page 3
Table of Contents......Page 5
Notes for the Reader......Page 7
Introduction......Page 9
I. Representations of finite groups......Page 27
1.1 Representations of groups......Page 28
1.2 Representations by unitary matrices, and complete reducibility of representations of finite groups......Page 33
1.3 Schur's lemma......Page 36
1.4 Orthogonality relations of characters......Page 38
1.5 Induced representations......Page 49
1.6 Product of representations......Page 55
1.7 Sharp permutation groups......Page 60
1.8 Notes to chapter I......Page 66
II. Association schemes......Page 68
2.1 Centralizer rings of permutation representations......Page 71
2.2 Association schemes......Page 78
2.3 Adjacency algebras (Bose-Mesner algebras), orthogonality relations of eigenmatrices, and Krein parameters......Page 84
2.4 Formulae for m_i......Page 97
2.5 The Kawada-Delsarte duality in C-algebras......Page 113
2.6 S-rings and permutation groups with regular subgroups......Page 130
2.7 An interpretation of the character table of a finite group......Page 139
2.8 Norton algebras and permutation groups......Page 149
2.9 Primitivity and imprimitivity of association schemes......Page 160
2.10 Tannaka-Krein duality for non-commutative association schemes......Page 179
2.11 Spherical functions of multiplicity-free permutation representations......Page 193
2.12 Notes to chapter II......Page 205
III. Distance-regular graphs and (P and Q)-polynomial association schemes......Page 211
3.1 Distance-regular graphs and P-polynomial association schemes......Page 214
3.2 The Hamming schemes H(n,q) and the Johnson schemes J(v,k)......Page 233
3.3 Moore graphs and related extremal cases......Page 251
3.4 Multiple P-polynomial structures......Page 264
3.5 (P and Q)-polynomial association schemes and theorem of Leonard......Page 284
3.6 The list of known (P and Q)-polynomial association schemes......Page 324
3.7 The rationality of eigenvalues of (P and Q)-polynomial schemes......Page 354
3.8 A review of the characterization problem of known (P and Q)-polynomial schemes by the parameters......Page 390
3.9 Notes to chapter III......Page 397
References......Page 409
Index......Page 442
Important notation......Page 448