Cyclage, catabolism, and the affine Hecke algebraAlgebra & Discrete Mathematics
|Jonah Blasiak, UC Berkeley
|Fri, Nov 21 2008, 2:10PM
It is classically known that the ring of coinvariants C[y_1, ..., y_n]/(e_1, ...,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. A subalgebra of this Hecke algebra has a cellular subquotient which is a q-analog of the ring of coinvariants and, conjecturally, has cellular subquotients that are q-analogs of the Garsia-Procesi modules. This viewpoint makes the appearance of cyclage in the combinatorial description of these modules transparent and has led to a new characterization of catabolizability.