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Wednesday, November 18, 2009

Algebra, Geometry and Combinatorics Seminar
12:00 pm   in 445 Altgeld Hall,  Wednesday, November 18, 2009
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Submitted by llpku.
Jonah Blasiak (University of Chicago)
Cyclage, catabolism, and the affine Hecke algebra
Abstract: 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. 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.