Bergeron Algebraic Combinatorics and Coinvariant Spaces

Erscheinungsjahr 2011
ISBN: 978-1-4398-6507-1
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 230 Seiten

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.

Zielgruppe

Graduate students in mathematics or non-specialist mathematicians.

Autoren/Hrsg.

Bergeron, Francois

Fachgebiete

• Mathematik | Informatik Mathematik Operations Research Graphentheorie

Inhaltsverzeichnis

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