A Combinatorial Formula for the Coefficient of q in Kazhdan–Lusztig Polynomials

Abstract

We propose a combinatorial interpretation of the coefficient of $q$ in Kazhdan–Lusztig polynomials and we prove it for finite simply-laced Weyl groups.

Combinatorial invariance for elementary intervals

“…In this section we introduce a refinement of the notion of a hypercube decomposition, which was introduced in [3]. Of particular importance will be various operations on "diamonds," some of which were discussed in [23]. Definition 3.1.…”

“…Other known cases include intervals in S n of length at most 8 [18,19] and all intervals in the rank-three affine Weyl group of type A 2 [9]. Patimo [23] has also shown that the coefficient of q in P u,v (q) is combinatorial for an arbitrary interval in S n .…”

Untitled

“…The combinatorial invariance conjecture is a central conjecture in the study of Bruhat intervals. The reader is referred to [Bre02] for more detail on known cases (see also [Dye87,Bre04,Inc06,Pat21,BLP21]). We do not discuss the various partial results towards the conjecture here, except to mention that it is known to hold for intervals starting at the identity [BCM06].…”

Towards combinatorial invariance for Kazhdan-Lusztig polynomials

“…The combinatorial invariance conjecture is a central conjecture in the study of Bruhat intervals. The reader is referred to [Bre04] for more detail on known cases (see also [Dye87,Bre04,Inc06,Pat21,BLP21]). We do not discuss the various partial results towards the conjecture here, except to mention that it is known to hold for intervals starting at the identity [BCM06].…”