On parabolic Kazhdan-Lusztig R-polynomials for the symmetric group

TL;DR

This paper advances the computation of parabolic R-polynomials for the symmetric group by introducing a new statistic and formulas for specific subsets, extending previous results and proposing a conjecture for broader cases.

Contribution

It introduces a new combinatorial statistic and provides explicit formulas for parabolic R-polynomials in certain cases, extending prior work by Brenti and proposing a conjecture for more general cases.

Findings

Derived a formula for J = S extbackslash extbraceleft s extsubscript{i-2}, s extsubscript{i-1}, s extsubscript{i} extbraceright.

Introduced a new permutation statistic for specific parabolic subsets.

Posed a conjecture for a broader class of parabolic R-polynomials.

Abstract

Parabolic $R$-polynomials were introduced by Deodhar as parabolic analogues of ordinary $R$-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic $R$-polynomials for the symmetric group. Let $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$, and let $S=\{s_i\,|\, 1\leq i\leq n-1\}$ be the generating set of $S_n$, where for $1\leq i\leq n-1$, $s_i$ is the adjacent transposition. For a subset $J\subseteq S$, let $(S_n)_J$ be the parabolic subgroup generated by $J$, and let $(S_n)^{J}$ be the set of minimal coset representatives for $S_n/(S_n)_J$. For $u\leq v\in (S_n)^J$ in the Bruhat order and $x\in \{q,-1\}$, let $R_{u,v}^{J,x}(q)$ denote the parabolic $R$-polynomial indexed by $u$ and $v$. Brenti found a formula for $R_{u,v}^{J,x}(q)$ when $J=S\setminus\{s_i\}$, and obtained an expression for $R_{u,v}^{J,x}(q)$ when $J=S\setminus\{s_{i-1},s_i\}$. We introduce a statistic on pairs of permutations in $(S_n)^J$ for $J=S\setminus\{s_{i-2},s_{i-1},s_i\}$. Then we give a formula for $R_{u,v}^{J,x}(q)$, where $J=S\setminus\{s_{i-2},s_{i-1},s_i\}$ and $i$ appears after $i-1$ in $v$. We also pose a conjecture for $R_{u,v}^{J,x}(q)$, where $J=S\setminus\{s_{k},s_{k+1},\ldots,s_i\}$ with $1\leq k\leq i\leq n-1$ and the elements $k+1,k+2,\ldots, i$ appear in increasing order in $v$.