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On parabolic Kazhdan-Lusztig \(R\)-polynomials for the symmetric group. (English) Zbl 1347.20012

Summary: 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 \(S=\{s_i\mid 1\leq i\leq n-1\}\) 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 \(J=S\setminus\{s_{i-1},s_i\}\). In this paper, we provide 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 \(v\). It should be noted that the condition that \(i\) appears after \(i-1\) in \(v\) is equivalent to that \(v\) is a permutation in \((S_n)^{S\setminus\{s_{i-2},s_i\}}\). 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 \(v\) is a permutation in \((S_n)^{S\setminus\{s_k,s_i\}}\).

MSC:

20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20C08 Hecke algebras and their representations
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References:

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