On the Kazhdan--Lusztig cells in type E8E_8

In 1979, Kazhdan and Lusztig introduced the notion of "cells" (left, right and two-sided) for a Coxeter group $W$, a concept with numerous applications in Lie theory and around. Here, we address algorithmic aspects of this theory for finite $W$ which are important in applications, e.g., run explicitly through all left cells, determine the values of Lusztig's \ba-function, identify the characters of left cell representations. The aim is to show how type $E_8$ (the largest group of exceptional type) can be handled systematically and efficiently, too. This allows us, for the first time, to solve some open questions in this case, including Kottwitz' conjecture on left cells and involutions. Further experiments suggest a characterisation of left cells, valid for any finite $W$, in terms of Lusztig's \ba-function and a slight modification of Vogan's generalized $\tau$-invariant.Comment: 21 pages; added Conjecture 6.9 and some minor correction

Combinatorial properties of Temperley Lieb algebras

We consider two families of polynomials that play the same role in the Temperley Lieb algebra of a Coxeter group as the Kazhdan Lusztig and R polynomials play in the Hecke algebra of the group. We study these polynomials from a combinatorial point of view. More precisely we obtain recursions, non recursive formulas, symmetry properties, and expressions for the constant terms, of these polynomials.Comment: This work is based on the author's doctoral dissertation, written under the direction of Prof. F. Brenti at the University of Rome "Tor Vergata