[Cherubino] Quaderni Elettronici del Seminario di Geometria Combinatoria 4E (Aprile 2005), 1--12
Università degli Studi di Roma "La Sapienza" - Dipartimento di Matematica

Federico INCITTI
Combinatorial invariance of Kazhdan-Lusztig polynomials for short intervals in the symmetric group
(Aprile 2005)

Abstract. The well-known combinatorial invariance conjecture states that, given a Coxeter group W, ordered by Bruhat order, and given two elements x,y in W, with x < y, the Kazhdan-Lusztig polynomial, or equivalently the R-polynomial, associated with (x,y) supposedly depends only on the poset structure of the interval [x,y].

In this paper we solve the conjecture for the first open cases, showing that it is true for intervals of length 5 and 6 in the symmetric group. The main tool is a pictorial way for describing the Bruhat order in the symmetric group, namely the diagram of a pair of permutations. It is shown how the diagram of (x,y) allows to get information about the poset structure of [x,y], and about the R-polynomial associated with (x,y). As a parallel result, we obtain expressions of the R-polynomials for some general classes of pairs of permutations.

Author:
Federico Incitti
Institut Mittag-Leffler
Auravägen 17
18260, Djursholm, Sweden
e-mail: incitti@math.kth.se
The following version is available: You might also want to have a look at the slides from F. Incitti's talk.