Reduced decompositions in Weyl groups
type: Item
instance of: scholarly article
title: Reduced decompositions in Weyl groups
zbMATH Open document ID: 0828.05064
DOI: 10.1016/0195-6698(95)90033-0
author: Witold Kraśkiewicz
published in: European Journal of Combinatorics
publication date: 1995-09-04T00:00:00Z
review text: Let \(R\) be a root system with fixed basis \(\Sigma\) and let \(W\) be the Weyl group of it. For any element \(w\in W\), let \(\Gamma_w\) be the set of inversions of \(w\), i.e. the set of positive roots \(\alpha\) such that \(w(\alpha)\) is negative. It is known that if \(w= s_1 s_2\dots s_k\) is a reduced decomposition in \(W\), with \(s_i\) being reflection with respect to \(\alpha_i\in \Sigma\), then \(\Gamma_w\) is equal to the set of \(\theta_i= s_k ssbk- 1\dots s_{i+ 1}(\alpha_i)\), \(i= 1, 2,\dots, k\). In this way parameterization of reduced decompositions of \(w\) by some linear orders of \(\Gamma_w\) is obtained. In the paper, linear orders of \(\Gamma_w\) are replaced by an equivalent notion of \(w\)-tableau and combinatorial characterization of standard \(w\)-tableaux, i.e. tableaux corresponding to reduced decompositions are given. For some special elements \(w\), \(w\)-tableaux are in natural one-to-one correspondence with standard Young tableaux which leads to explicit formulae for the number of reduced decompositions of \(w\). If \(W\) is the symmetric group \(S_n\) and \(w\) is any element of \(W\) then slight modification of tableau constructions of irreducible representations of the symmetric groups allows to define a linear action of the symmetric group \(S_{\ell(w)}\) on the space spanned by standard \(w\)-tableaux. It is an interpretation of Stanley's result [\textit{R. P. Stanley}, Eur. J. Comb. 5, 359-372 (1984; Zbl 0587.20002)] on the number of reduced decompositions of permutations in the framework of representation theory.
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 773832
zbMATH Keywords: root system
zbMATH Keywords: Weyl group
zbMATH Keywords: reduced decomposition
zbMATH Keywords: Young tableaux
zbMATH Keywords: symmetric group
zbMATH Keywords: irreducible representations
MaRDI profile type: Publication
full work available at URL: 0195-6698(95)90033-0
OpenAlex ID: W2118640839
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description: scientific article; zbMATH DE number 773832