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

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• 作者：Neil J.Y. Fan^a ; ^{fan@scu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Peter L. Guo^b ; ^{lguo@nankai.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Grace L.D. Zhang^b ; ^{zhld@mail.nankai.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：20F55 ; 05E99
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：221
• 期：1
• 页码：237-250
• 全文大小：359 K

文摘

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 55473a980a598f8f5a0fa" title="Click to view the MathML source">S_n$S_n$ be the symmetric group on {1,2,…,n}$\{1,2,\dots ,n\}$, and let 682e4443ee8e8752445f9fa0">$S=\{s_i|1\le i\le n-1\}$ be the generating set of 55473a980a598f8f5a0fa" title="Click to view the MathML source">S_n$S_n$, where for 684f505038d0ef09931" title="Click to view the MathML source">1≤i≤n−1$1\le i\le n-1$, s_i$s_i$ is the adjacent transposition. For a subset 55fdba384fd342a67cad6a5" title="Click to view the MathML source">J⊆S$J\subseteq S$, let 6828c6f758" title="Click to view the MathML source">(S_n)_J${(S_n)}_J$ be the parabolic subgroup generated by J  , and let (S_n)^J${(S_n)}^J$ be the set of minimal coset representatives for 5507c05db6be8f6fad98d280ba2c8dc" title="Click to view the MathML source">S_n/(S_n)_J$S_n/{(S_n)}_J$. For u≤v∈(S_n)^J$u\le v\in {(S_n)}^J$ in the Bruhat order and x∈{q,−1}$x\in \{q,-1\}$, 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∖{s_i}$J=S\setminus \{s_i\}$, and obtained an expression for $R_{u,v}^{J,x}(q)$ when J=S∖{s_i−1,s_i}$J=S\setminus \{s_{i-1},s_i\}$. In this paper, we provide a formula for $R_{u,v}^{J,x}(q)$, where J=S∖{s_i−2,s_i−1,s_i}$J=S\setminus \{s_{i-2},s_{i-1},s_i\}$ and i   appears after i−1$i-1$ in v. It should be noted that the condition that i   appears after i−1$i-1$ in v is equivalent to that v   is a permutation in (S_n)^{S∖{s_i−2,s_i}}${(S_n)}^{S\setminus \{s_{i-2},s_i\}}$. We also pose a conjecture for $R_{u,v}^{J,x}(q)$, where 55e3108ca86b627d0312349c4214" title="Click to view the MathML source">J=S∖{s_k,s_k+1,…,s_i}$J=S\setminus \{s_k,s_{k+1},\dots ,s_i\}$ with 1≤k≤i≤n−1$1\le k\le i\le n-1$ and v   is a permutation in (S_n)^{S∖{s_k,s_i}}${(S_n)}^{S\setminus \{s_k,s_i\}}$.