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Cyclage, catabolism, and the affine Hecke algebra
Algebra & Discrete Mathematics| Speaker: | Jonah Blasiak, UC Berkeley |
| Location: | 2112 MSB |
| Start time: | Fri, Nov 21 2008, 2:10PM |
Description
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.