The University of Michigan Combinatorics Seminar


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 qanalog of the ring of coinvariants and, further, that this subquotient has cellular quotients which are qanalogs of the GarsiaProcesi modules. This cellular picture gives a clear explanation of the appearance of cyclage and catabolism in the combinatorial description of these modules. 