Algebraic Combinatorics and Coinvariant Spaces: 1st Edition (Hardback) book cover

Algebraic Combinatorics and Coinvariant Spaces

1st Edition

By Francois Bergeron

A K Peters/CRC Press

230 pages

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pub: 2009-07-06
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Description

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.

Table of Contents

Combinatorial Objects

Permutations

Monomials

Diagrams

Partial Orders on Vectors

Young Diagrams

Partitions

The Young Lattice

Compositions

Words

Some Tableau Combinatorics

Tableaux

Insertion in Words and Tableaux

Jeu de Taquin

The RSK Correspondence

Viennot’s Shadows

Charge and Cocharge

Invariant Theory

Polynomial Ring

Root Systems

Coxeter Groups

Invariant and Skew-Invariant Polynomials

Symmetric Polynomials and Functions

The Fundamental Theorem of Symmetric Functions

More Basic Identities

Plethystic Substitutions

Antisymmetric Polynomials

Schur and Quasisymmetric Functions

A Combinatorial Approach

Formulas Derived from Tableau Combinatorics

Dual Basis and Cauchy Kernel

Transition Matrices

Jacobi–Trudi Determinants

Proof of the Hook Length Formula

The Littlewood–Richardson Rule

Schur-Positivity

Poset Partitions

Quasisymmetric Functions

Multiplicative Structure Constants

r-Quasisymmetric Polynomials

Representation Theory

Basic Representation Theory

Characters

Special Representations

Action of Sn on Bijective Tableaux

Irreducible Representations of Sn

Frobenius Transform

Restriction from Sn to Sn−1

Polynomial Representations of GL (V)

Schur–Weyl Duality

Species Theory

Species of Structures

Generating Series

The Calculus of Species

Vertebrates and Rooted Trees

Generic Lagrange Inversion

Tensorial Species

Polynomial Functors

Commutative Algebra

Ideals and Varieties

Gröbner Basis

Graded Algebras

The Cohen–Macaulay Property

Coinvariant Spaces

Coinvariant Spaces

Harmonic Polynomials

Regular Point Orbits

Symmetric Group Harmonics

Graded Frobenius Characteristic of the Sn-Coinvariant

Space

Generalization to Line Diagrams

Tensorial Square

Macdonald Functions

Macdonald’s Original Definition

Renormalization

Basic Formulas

q, t-Kostka

A Combinatorial Approach

Nabla Operator

Diagonal Coinvariant Spaces

Garsia–Haiman Representations

Generalization to Diagrams

Punctured Partition Diagrams

Intersections of Garsia–Haiman Modules

Diagonal Harmonics

Specialization of Parameters

Coinvariant-Like Spaces

Operator-Closed Spaces

Quasisymmetric Modulo Symmetric

Super-Coinvariant Space

More Open Questions

Formulary

Some q-Identities

Partitions and Tableaux

Symmetric and Antisymmetric Functions

Integral Form Macdonald Functions

Some Specific Values

Subject Categories

BISAC Subject Codes/Headings:
MAT036000
MATHEMATICS / Combinatorics