Group Theory: Cyclage, catabolism and the affine Hecke algebra
by J. Blasiak (U. Chicago (CS)) in E 203
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.