A combinatorial formula for Kazhdan-Lusztig polynomials

Abstract

In their fundamental paper [14] 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 (see, e.g., [12], Chap. 7). These polynomials are intimately related to the Bruhat order of W and to the algebraic geometry of Schubert varieties, and have proven to be of fundamental importance in representation theory. Our aim in this work is to give a simple, nonrecursive, combinatorial formula for any Kazhdan-Lusztig polynomial of any Coxeter group (Theorem 4.1), and to study some consequences of it. The main idea involved in the proof and statement of this formula is that of extending the concept of the Rpolynomial (see, e.g., [12], w to any (finite) multichain of W (so that, for multichains of length 1, one obtains, apart from sign, the usual R-polynomials). Once this has been done, then the Kazhdan-Lusztig polynomial of a pair u, v turns out to be just the sum, over all multichains from u to v, of the corresponding (generalized) R-polynomials. The R-polynomial of a multichain can be readily defined, and computed from the ordinary R-polynomials (see (9), (10), and (11)). Since several combinatorial formulas and interpretations are known for these polynomials (see, e.g., [5, 10], and, for the case of symmetric groups, [3]) and simple recurrences exist for them, we feel that this formula is a significant step forward in the understanding of the KazhdanLusztig polynomials. Though combinatorial formulas for Kazhdan-Lusztig polynomials have appeared before in the literature (see, e.g., [16, 21, 8, 6]), none of them hold in complete generality.