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. Some Tableau Combinatorics. Invariant Theory. Schur and Quasisymmetric Functions. Representation Theory. Species Theory. Commutative Algebra. Coinvariant Spaces. Macdonald Functions. Diagonal Coinvariant Spaces. Coinvariant-Like Spaces. Formulary