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 S_n$S_n$ be the symmetric group on {1,2,…,n}$\{1,2,\dots ,n\}$, and let 45f9fa0">$S=\{s_i|1\le i\le n\&<font\; color="red">minus</font>;1\}$ be the generating set of S_n$S_n$, where for 05038d0ef09931" title="Click to view the MathML source">1≤i≤n&minus;1$1\le i\le n\&<font\; color="red">minus</font>;1$, s_i$s_i$ is the adjacent transposition. For a subset J⊆S$J\subseteq S$, let (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 05db6be8f6fad98d280ba2c8dc" 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 052589" title="Click to view the MathML source">x∈{q,&minus;1}$x\in \{q,\&<font\; color="red">minus</font>;1\}$, let 456">$R_{u,v}^{J,x}(q)$ denote the parabolic R-polynomial indexed by u and v  . Brenti found a formula for 456">$R_{u,v}^{J,x}(q)$ when 45f302c760a24ab3c" title="Click to view the MathML source">J=S∖{s_i}$J=S\setminus \{s_i\}$, and obtained an expression for 456">$R_{u,v}^{J,x}(q)$ when 45ad785893c3a846c30" title="Click to view the MathML source">J=S∖{s_i&minus;1,s_i}$J=S\setminus \{s_{i\&<font\; color="red">minus</font>;1},s_i\}$. In this paper, we provide a formula for 456">$R_{u,v}^{J,x}(q)$, where J=S∖{s_i&minus;2,s_i&minus;1,s_i}$J=S\setminus \{s_{i\&<font\; color="red">minus</font>;2},s_{i\&<font\; color="red">minus</font>;1},s_i\}$ and i   appears after i&minus;1$i\&<font\; color="red">minus</font>;1$ in v. It should be noted that the condition that i   appears after i&minus;1$i\&<font\; color="red">minus</font>;1$ in v is equivalent to that v   is a permutation in (S_n)^{S∖{s_i&minus;2,s_i}}${(S_n)}^{S\setminus \{s_{i\&<font\; color="red">minus</font>;2},s_i\}}$. We also pose a conjecture for 456">$R_{u,v}^{J,x}(q)$, where 05" class="mathmlsrc">05.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=745655e3108ca86b627d0312349c4214" title="Click to view the MathML source">J=S∖{s_k,s_k+1,…,s_i} with 1≤k≤i≤n&minus;1$1\le k\le i\le n\&<font\; color="red">minus</font>;1$ and v   is a permutation in (S_n)^{S∖{s_k,s_i}}${(S_n)}^{S\setminus \{s_k,s_i\}}$.