Algebraic Combinatorics and Coinvariant Spaces

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.

Francois Bergeron is one of the founding members of the Montreal school of combinatorics which developed the Theory of Species of Structures as a comprehensive tool for enumerative combinatorics. He currently holds the position of professor at the Universite de Quebec a Montreal, where he is also the director of the research center: Laboratoire de Combinatoire et Informatique Mathematique. He received his Ph.D. in Mathematics from the Universite de Montreal in 1987. Francois Bergeron has published numerous articles in the areas of algebraic combinatorics and enumerative combinatorics. He is one of the co-author of the book Combinatorial Species and Tree-Like Structures, which is the main reference on the subject. He is also an ardent proponent of the use of computer algebra systems and experimental mathematics as a support for theoretical research.