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 80&_mathId=si1.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=e8e91fb8f8655473a980a598f8f5a0fa" title="Click to view the MathML source">S_n$S_n$ be the symmetric group on 80&_mathId=si2.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=08da7bb8ac97a48b512fec065e5dd38c" title="Click to view the MathML source">{1,2,…,n}$\{1,2,\dots ,n\}$, and let 80&_mathId=si3.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=29c095a2682e4443ee8e8752445f9fa0">80-si3.gif"> be the generating set of 80&_mathId=si1.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=e8e91fb8f8655473a980a598f8f5a0fa" title="Click to view the MathML source">S_n$S_n$, where for 80&_mathId=si4.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=1da6fb7ba18a0684f505038d0ef09931" title="Click to view the MathML source">1≤i≤n−1$1\le i\le n-1$, 80&_mathId=si5.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=f8ae08d1cf14037e3bce61a2a7eef932" title="Click to view the MathML source">s_i$s_i$ is the adjacent transposition. For a subset 80&_mathId=si6.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=0d13c888855fdba384fd342a67cad6a5" title="Click to view the MathML source">J⊆S$J\subseteq S$, let 80&_mathId=si7.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=a1c2a678d7721e6f697f546828c6f758" title="Click to view the MathML source">(S_n)_J${(S_n)}_J$ be the parabolic subgroup generated by J  , and let 80&_mathId=si8.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=aa1cf939baf69187fd7813e6cf431e93" title="Click to view the MathML source">(S_n)^J${(S_n)}^J$ be the set of minimal coset representatives for 80&_mathId=si9.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=95507c05db6be8f6fad98d280ba2c8dc" title="Click to view the MathML source">S_n/(S_n)_J$S_n/{(S_n)}_J$. For 80&_mathId=si10.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=778cc6535819bfcce611642a1f7b5f38" title="Click to view the MathML source">u≤v∈(S_n)^J$u\le v\in {(S_n)}^J$ in the Bruhat order and 80&_mathId=si11.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=0a21dd08c38848463b399d953e052589" title="Click to view the MathML source">x∈{q,−1}$x\in \{q,-1\}$, let 80&_mathId=si12.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=79306b58c8fcac8a76e461543c2f6456">80-si12.gif">$R_{u,v}^{J,x}(q)$ denote the parabolic R-polynomial indexed by u and v  . Brenti found a formula for 80&_mathId=si12.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=79306b58c8fcac8a76e461543c2f6456">80-si12.gif">$R_{u,v}^{J,x}(q)$ when 80&_mathId=si13.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=2a61efd2d8af17b45f302c760a24ab3c" title="Click to view the MathML source">J=S∖{s_i}$J=S\setminus \{s_i\}$, and obtained an expression for 80&_mathId=si12.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=79306b58c8fcac8a76e461543c2f6456">80-si12.gif">$R_{u,v}^{J,x}(q)$ when 80&_mathId=si14.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=4c308a849ad9145ad785893c3a846c30" title="Click to view the MathML source">J=S∖{s_i−1,s_i}$J=S\setminus \{s_{i-1},s_i\}$. In this paper, we provide a formula for 80&_mathId=si12.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=79306b58c8fcac8a76e461543c2f6456">80-si12.gif">$R_{u,v}^{J,x}(q)$, where 80&_mathId=si102.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=24036d5f39154e2f92bde731f2746b62" title="Click to view the MathML source">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 80&_mathId=si16.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=3129bf4c17af4fca3df6d8c5a241c5e3" title="Click to view the MathML source">i−1$i-1$ in v. It should be noted that the condition that i   appears after 80&_mathId=si16.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=3129bf4c17af4fca3df6d8c5a241c5e3" title="Click to view the MathML source">i−1$i-1$ in v is equivalent to that v   is a permutation in 80&_mathId=si103.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=8b11bd77f007398a8c6407e30f0e37e0" title="Click to view the MathML source">(S_n)^{S∖{s_i−2,s_i}}${(S_n)}^{S\setminus \{s_{i-2},s_i\}}$. We also pose a conjecture for 80&_mathId=si12.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=79306b58c8fcac8a76e461543c2f6456">80-si12.gif">$R_{u,v}^{J,x}(q)$, where 80&_mathId=si105.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}$J=S\setminus \{s_k,s_{k+1},\dots ,s_i\}$ with 80&_mathId=si19.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=d09c021608a5a78c967704c6377caae8" title="Click to view the MathML source">1≤k≤i≤n−1$1\le k\le i\le n-1$ and v   is a permutation in 80&_mathId=si20.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=c99bd72a4a3c29196cb8bf4f62f83f5c" title="Click to view the MathML source">(S_n)^{S∖{s_k,s_i}}${(S_n)}^{S\setminus \{s_k,s_i\}}$.