, Volume 36, Issue 1, pp 95-119

A combinatorial setting for questions in Kazhdan — Lusztig theory

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Abstract

Let(W, S) be a Coxeter group. Let s 1 ... s k be a reduced expression for an element y in W. A combinatorial setting involving subexpressions of this reduced expression is developed. This leads to the notion of good elements. It is proved that all elements in a group where the coefficients of Kazhdan — Lusztig polynomials are non-negative are good. If y is good then an algorithm is developed to compute these polynomials in a very efficient way. It is further proved that in these cases, the coefficients of these polynomials can be identified as sizes of certain subsets of subexpressions thereby providing an explicit setting for various questions regarding these polynomials and related topics. Similar results are obtained for the so-called parabolic case.

Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthday
Partially supported by NSF grant No. DMS 8502310.