Special matchings and Kazhdan–Lusztig polynomials

Abstract

In 1979 Kazhdan and Lusztig defined, for every Coxeter group W , a family of polynomials, indexed by pairs of elements of W , which have become known as the Kazhdan–Lusztig polynomials of W , and which have proven to be of importance in several areas of mathematics. In this paper, we show that the combinatorial concept of a special matching plays a fundamental role in the computation of these polynomials. Our results also imply, and generalize, the recent one in [Adv. in Math. 180 (2003) 146–175] on the combinatorial invariance of Kazhdan–Lusztig polynomials.