Written for graduate students in mathematics or non-specialist mathematicians who wish to learn the basics about some of the most important current research in the field, this book provides an intensive, yet accessible, introduction to the subject of algebraic combinatorics. After recalling basic notions of combinatorics, representation theory, and some commutative algebra, the main material provides links between the study of coinvariant or diagonally coinvariant spaces and the study of Macdonald polynomials and related operators. This gives rise to a large number of combinatorial questions relating to objects counted by familiar numbers such as the factorials, Catalan numbers, and the number of Cayley trees or parking functions. The author offers ideas for extending the theory to other families of finite Coxeter groups, besides permutation groups.
Author(s): François Bergeron
Series: CMS Treatises in Mathematics/ Traites de Mathematiques de la SMC
Publisher: AK Peters
Year: 2009
Language: English
Commentary: index is missing
Pages: 226
Tags: Математика;Дискретная математика;Комбинаторика;
Contents......Page 6
Introduction......Page 10
1.1 Permutations......Page 14
1.2 Monomials......Page 18
1.3 Diagrams......Page 19
1.4 Partial Orders on Vectors......Page 21
1.5 Young Diagrams......Page 23
1.6 Partitions......Page 24
1.7 The Young Lattice......Page 29
1.8 Compositions......Page 35
1.9 Words......Page 36
2.1 Tableaux......Page 40
2.2 Insertion in Words and Tableaux......Page 44
2.3 Jeu de Taquin......Page 45
2.4 The RSK Correspondence......Page 46
2.5 Viennot's Shadows......Page 49
2.6 Charge and Cocharge......Page 52
3.1 Polynomial Ring......Page 54
3.3 Coxeter Groups......Page 60
3.4 Invariant and Skew-Invariant Polynomials......Page 61
3.5 Symmetric Polynomials and Functions......Page 64
3.7 More Basic Identities......Page 69
3.8 Plethystic Substitutions......Page 71
3.9 Antisymmetric Polynomials......Page 74
4.1 A Combinatorial Approach......Page 78
4.2 Formulas Derived from Tableau Combinatorics......Page 80
4.3 Dual Basis and Cauchy Kernel......Page 82
4.4 Transition Matrices......Page 84
4.5 Jacobi–Trudi Determinants......Page 85
4.6 Proof of the Hook Length Formula......Page 89
4.7 The Littlewood–Richardson Rule......Page 92
4.8 Schur-Positivity......Page 94
4.9 Poset Partitions......Page 96
4.10 Quasisymmetric Functions......Page 97
4.11 Multiplicative Structure Constants......Page 101
4.12 r-Quasisymmetric Polynomials......Page 103
5.1 Basic Representation Theory......Page 104
5.2 Characters......Page 105
5.3 Special Representations......Page 106
5.4 Action of S n on Bijective Tableaux......Page 107
5.5 Irreducible Representations of S n......Page 109
5.6 Frobenius Transform......Page 110
5.8 Polynomial Representations of GL (V)......Page 114
5.9 Schur–Weyl Duality......Page 116
6.1 Species of Structures......Page 118
6.2 Generating Series......Page 120
6.3 The Calculus of Species......Page 121
6.4 Vertebrates and Rooted Trees......Page 126
6.5 Generic Lagrange Inversion......Page 127
6.6 Tensorial Species......Page 129
6.7 Polynomial Functors......Page 132
7.1 Ideals and Varieties......Page 134
7.2 Gröbner Basis......Page 135
7.3 Graded Algebras......Page 137
7.4 The Cohen–Macaulay Property......Page 139
8.1 Coinvariant Spaces......Page 140
8.2 Harmonic Polynomials......Page 142
8.3 Regular Point Orbits......Page 143
8.4 Symmetric Group Harmonics......Page 144
8.5 Graded Frobenius Characteristic of the Sn-Coinvariant Space......Page 145
8.6 Generalization to Line Diagrams......Page 146
8.7 Tensorial Square......Page 151
9.1 Macdonald's Original Definition......Page 156
9.2 Renormalization......Page 158
9.3 Basic Formulas......Page 160
9.4 q, t-Kostka......Page 163
9.5 A Combinatorial Approach......Page 164
9.6 Nabla Operator......Page 167
10.1 Garsia–Haiman Representations......Page 170
10.2 Generalization to Diagrams......Page 173
10.3 Punctured Partition Diagrams......Page 176
10.4 Intersections of Garsia–Haiman Modules......Page 180
10.5 Diagonal Harmonics......Page 181
10.6 Specialization of Parameters......Page 185
11.1 Operator-Closed Spaces......Page 192
11.2 Quasisymmetric Modulo Symmetric......Page 194
11.3 Super-Coinvariant Space......Page 198
11.4 More Open Questions......Page 203
A.1 Some q-Identities......Page 206
A.2 Partitions and Tableaux......Page 207
A.3 Symmetric and Antisymmetric Functions......Page 209
A.4 Integral Form Macdonald Functions......Page 212
A.5 Some Specific Values......Page 214
Bibliography......Page 218
