University of Chicago
Department of Mathematics

It is classically known that the ring of coinvariants $\CC[y_1,\ldots,y_n]/(e_1, \ldots,e_n)$, thought of as an $\S_n$-module with $\S_n$ acting by permuting the variables, is a graded version of the regular representation of $\S_n$. However, how a decomposition of the coinvariants into irreducibles is compatible with its ring structure remains a mystery. In particular, there are difficult combinatorial conjectures for the graded characters of certain subquotients of this ring. We describe a promising approach to understanding such subquotients using the canonical basis of the extended affine Hecke algebra. We show that a subalgebra of this Hecke algebra has a cellular subquotient which is a $q$-analog of the ring of coinvariants and, further, that this subquotient has cellular quotients which are $q$-analogs of the Garsia-Procesi modules. This cellular picture gives a clear explanation of the appearance of cyclage and catabolism in the combinatorial description of these modules.

If $k$ is a field, $G$ is a finite group, and $H$ is a subgroup of $G$, and $kG^H$ is definied to be the centralizer in $kG$ of $kH$. We will explore the problems of finding the simple $kG^H$-modules, the blocks of $kG^H$, and the decomposition matrix for $kG^H$, concentrating on the case when $G=S_n$ and $H=S_l$ with $l