Lattice Paths and Kazhdan-lusztig Polynomials

In their fundamental paper [18] 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., [17], Chap. 7). These polynomials are intimately related to the Bruhat order of W and to the geometry of Schubert varieties, and have proven to be of fundamental importance in representation theory. The purpose of this paper is to present a new non-recursive combinatorial formula for these polynomials. More precisely, we show that each directed path in the Bruhat graph of W has a naturally associated set of lattice paths with the property that the Kazhdan-Lusztig polynomial of u, v is the sum, over all the lattice paths associated to all the paths going from u to v, of (−1)Γ≥0+d+(Γ)q(l(v)−l(u)+Γ(l(Γ)))/2 where Γ≥0, d+(Γ), and Γ(l(Γ)) are three natural statistics on the lattice path. We believe that this formula is the most explicit non-recursive formula known for the Kazhdan-Lusztig polynomials which holds in complete generality. The organization of the paper is as follows. In section 3 we define and study the R-polynomial of a chain. This polynomial reduces to the usual R-polynomial for a chain of length one and is fundamental in all that follows. In section 4 we show that the antisymmetrization of the Kazhdan-Lusztig polynomial of two elements u, v of W equals the alternating sum, over all the chains going from u to v, of the corresponding R-polynomials. In section 5 we derive some consequences of this result which, although not needed in what follows, are of independent interest. More precisely, we derive explicit formulas for each coefficient of any Kazhdan-Lusztig polynomial, and comparing these to known formulas we obtain some new identities for the R-polynomials. In section 6 we introduce and study a family of polynomials, indexed by sequences of positive integers, which is needed in the proof of our main result. These polynomials are independent of W , are easily computable by simple recursions, and can be interpreted as counting certain lattice paths. In section 7 we prove our main theorem, which is obtained by combining the results in sections 4 and 6. This expresses the Kazhdan-Lusztig polynomial of two elements u, v in W as