Admissable W-graphs

A W-graph is a combinatorial structure that encodes a matrix representation of a Weyl group W, or more generally, a matrix representation for the Iwahori-Hecke algebra associated to W. Of special interest are the W-graphs that encode the action of the Hecke algebra on its Kazhdan-Lusztig basis, as well as the action on individual cells. Knowing the W-graph allows easy computation of the Kazhdan-Lusztig polynomials. In this talk we will isolate a few basic features common to the W-graphs in Kazhdan-Lusztig theory (thereby creating the class of "admissible" W-graphs), and explain how these features still manage to capture a large part of what is essential. For example, it turns out that there are only finitely many admissible W-cells (i.e., strongly connected W-graphs), and we are close to being able to determine the full Kazhdan-Lusztig W-graph via this approach.