The University of Michigan Combinatorics Seminar


Abstract 

A Wgraph is a weighted directed graph that encodes certain actions of a Coxeter group W or the associated IwahoriHecke algebra H(W). It is admissible if it is bipartite and has nonnegative integer edge weights that satisfy a simple symmetry condition. Of particular interest are the admissible Wgraphs and WxWgraphs that encode the onesided and twosided actions of the standard generators on the KazhdanLusztig basis of H(W), as well as the strongly connected components of these graphsthe latter are the socalled KazhdanLusztig cells. In this talk, we will report on progress toward understanding the structure and classification of admissible Wgraphs. An interesting recent development is the classification of all admissible W_{1}xW_{2}cells, where W_{1} and W_{2} both have rank two. This amounts to classifying pairs of simplylaced Cartan matrices of the same rank that commute and satisfy a simple parity condition. It turns out that there are 5 infinite families of such Cartan pairs (up to isomorphism), as well as 8 exceptional pairs whose ranks range from 12 to 32. 