The University of Michigan Combinatorics Seminar
A W-graph is a weighted directed graph that encodes certain actions of a Coxeter group W or the associated Iwahori-Hecke 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 W-graphs and WxW-graphs that encode the one-sided and two-sided actions of the standard generators on the Kazhdan-Lusztig basis of H(W), as well as the strongly connected components of these graphs---the latter are the so-called Kazhdan-Lusztig cells.
In this talk, we will report on progress toward understanding the structure and classification of admissible W-graphs. An interesting recent development is the classification of all admissible W1xW2-cells, where W1 and W2 both have rank two. This amounts to classifying pairs of simply-laced 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.