On Kazhdan--Lusztig basis elements having no reversal factorization

TL;DR

This paper investigates Kazhdan--Lusztig basis elements in the symmetric group, characterizing those that cannot be factorized into maximal parabolic elements, and explores combinatorial interpretations of related polynomials.

Contribution

It extends previous results to identify permutations with no such factorization, providing new insights into the structure of Kazhdan--Lusztig basis elements.

Findings

Characterizes permutations with no factorization into maximal parabolic elements.

Provides cancellation-free combinatorial interpretations of Kazhdan--Lusztig polynomials.

Describes a set of permutations that admit no such factorization.

Abstract

For $w$ in the symmetric group $S_n$, let $\widetilde C_w$ be the corresponding modified, signless Kazhdan--Lusztig basis element of the type-$A$ Hecke algebra $H_n(q)$. An extension [Ann. Comb. 25, no. 3 (2021) pp. 757--787] of a result of Deodhar [Geom. Dedicata 36, (1990) pp. 95--119] implies that any factorization of the form \begin{equation*} \widetilde C_w = \frac1{f(q)} \widetilde C_{v^{(1)}} \cdots \widetilde C_{v^{(r)}}, \end{equation*} with $v^{(1)},\dotsc,v^{(r)}$ maximal elements of parabolic subgroups of $S_n$ and $f(q) \in \mathbb N[q]$ depending on these, provides cancellation-free combinatorial interpretations of the polynomials $\{P_{v,w}(q) \,|\, v \in S_n \}$ appearing in the expansion $\sum_v P_{v,w}(q) T_v$ of $\widetilde C_w$ in terms of the natural basis $\{ T_v \,|\, v \in S_n \}$ of $H_n(q)$. While the set of permutations $w \in S_n$ admitting such a factorization of $\widetilde C_w$ has not yet been characterized, we apply a result of Gaetz -- Gao [Adv. Math. 457 (2024) Paper No. 109941] to describe a set admitting no such factorization.