Robinson-Schensted correspondence and left cells
type: Item
instance of: scholarly article
zbMATH Open document ID: 0986.05097
author: Susumu Ariki
publication date: 2002-06-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9910117
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 1642280
zbMATH Keywords: Robinson-Schensted correspondence
zbMATH Keywords: Kazhdan-Lusztig theorem
zbMATH Keywords: symmetric group
zbMATH Keywords: Bruhat order
zbMATH Keywords: tableaux
MaRDI profile type: Publication
arXiv ID: math/9910117
Recommended article: Note on the minimal permutations
Recommended article: Q4252966
Recommended article: Relations between Young's natural and the Kazhdan-Lusztig representations of \(S_ n\)
Recommended article: Proof of two conjectures of Brenti and Simion on Kazhdan-Lusztig polynomials
Recommended article: Minimal and maximal elements in two-sided cells of \(S_n\) and Robinson-Schensted correspondence
Recommended article: A formula for certain inverse Kazhdan--Lusztig polynomials in \(S_{n}\)
Recommended article: Construction de polynômes de Kazhdan-Lusztig arbitraires
Recommended article: A combinatorial setting for questions in Kazhdan-Lusztig theory
Recommended article: The leading coefficient of certain Kazhdan-Lusztig polynomials of the permutation group \({\mathfrak S}_n\).
Recommended article: qRSt: A Probabilistic Robinson–Schensted Correspondence for Macdonald Polynomials
title: Robinson-Schensted correspondence and left cells
review text: The Robinson-Schensted correspondence associates with any permutation \(w\) of the symmetric group \(S_n\) a pair of standard tableaux \((P(w),Q(w))\) called, respectively, the P-symbol and Q-symbol of \(w\). The Kazhdan-Lusztig polynomials \(P_{y,w}(q)\in{\mathbb Z}[q]\), where \(y\leq w\) with respect to the Bruhat order in \(S_n\), are defined by the equations NEWLINE\[NEWLINEC_w=\sum_{y\leq w}(-1)^{l(w)-l(y)}q^{l(w)/2-l(y)}P_{y,w}(q^{-1})T_y =\sum_{y\leq w}(-1)^{l(w)-l(y)}q^{l(w)/2+l(y)}P_{y,w}(q)T^{-1}_{y^{-1}},NEWLINE\]NEWLINE \(P_{w,w}(q)=1\) and \(\text{deg}P_{y,w}(q)\leq (l(w)-l(y)-1)/2\) for \(y<w\). Here \(l(w)\) is the length of a reduced expression of \(w\) with respect to the transpositions \(s_i=(i,i+1)\) and \(T_y\) is the element of the Hecke algebra of \(S_n\) related with \(y\). Let \(\mu(y|w)\not=0\) mean that either \(y<w\) and the coefficient \(\mu(y,w)\) of \(q^{(l(w)-l(y)-1)/2}\) in \(P_{y,w}(q)\) is nonzero or \(\mu(w,y)\not=0\) and \(y>w\), and let \({\mathcal L}(w)=\{s_j\mid s_jw<w\}\). One defines \(y\leq_L w\) if there exists a sequence \(y=x_1,x_2,\ldots,x_r=w\) such that \({\mathcal L}(x_i)\not\subset {\mathcal L}(x_{i+1})\) and \(\mu(x_i|x_{i+1})\not=0\) for all \(i=1,2,\ldots,r-1\). The permutations \(y\) and \(w\) belong to the same left cell of \(S_n\) if both \(y\leq_L w\) and \(w\leq_L y\) hold. The theorem of Kazhdan-Lusztig states that two permutations \(y\) and \(w\) belong to the same left cell of \(S_n\) if and only if they share a common Q-symbol. This theorem is not explicitly stated in the text of \textit{D. Kazhdan} and \textit{G. Lusztig} [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] but follows from there with some additional arguments from \textit{A. Joseph} [Lect. Notes Math. 587, 102-118 (1977; Zbl 0374.17004)]. The original proof of Kazhdan and Lusztig is quite complicated. There are other proofs; see \textit{A. M. Garsia} and \textit{T. J. McLarnan} [Adv. Math. 69, No. 1, 32-92 (1988; Zbl 0657.20014)] and \te…
reviewed by: Vesselin Drensky
description: scientific article; zbMATH DE number 1642280