Kazhdan--Lusztig polynomials for maximally-clustered hexagon-avoiding permutations

TL;DR

This paper introduces a combinatorial method to compute Kazhdan--Lusztig polynomials for a specific class of permutations in type A, characterized by pattern avoidance and maximal clustering, simplifying previous recursive approaches.

Contribution

It provides a non-recursive, combinatorial description for calculating Kazhdan--Lusztig polynomials for maximally-clustered hexagon-avoiding permutations, expanding understanding of their structure.

Findings

Non-recursive description of admissible sets for these permutations

Characterization of permutations by pattern avoidance

Application of heaps to permutation pattern analysis

Abstract

We provide a non-recursive description for the bounded admissible sets of masks used by Deodhar's algorithm to calculate the Kazhdan--Lusztig polynomials $P_{x,w}(q)$ of type $A$, in the case when $w$ is hexagon avoiding and maximally clustered. This yields a combinatorial description of the Kazhdan--Lusztig basis elements of the Hecke algebra associated to such permutations $w$. The maximally-clustered hexagon-avoiding elements are characterized by avoiding the seven classical permutation patterns $\{3421, 4312, 4321, 46718235, 46781235, 56718234, 56781234\}$. We also briefly discuss the application of heaps to permutation pattern characterization.