Note on a theorem of Eng
type: Item
instance of: scholarly article
title: Note on a theorem of Eng
zbMATH Open document ID: 1022.20017
author: Victor Reiner
published in: Annals of Combinatorics
publication date: 2003-03-25T00:00:00Z
review text: Let \((W,S)\) be a finite Coxeter system, \(J\) a subset of \(S\), \(W^J\) the distinguished left coset representatives for the parabolic subgroup \(W_J\) in \(W\), and \(w_0\) the longest element of \(W\). \textit{O. D. Eng} [J. Algebr. Comb. 13, No.1, 29-40 (2001; Zbl 1021.20031)], proved the following theorem. \[ [\sum_{w\in W^J}q^{l(w)}]_{q=-1}=\#\{w\in W^J;\;w_0wW_J=wW_J\}. \] Eng proved this by appeal to the classification of finite irreducible Coxeter systems, and asked for a case-free proof. In this paper, the author gives such a proof via geometry in the case where \((W,S)\) is a Weyl group.
reviewed by: Chen Chengdong
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 05E15
zbMATH DE Number: 1887494
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Poincaré polynomials
zbMATH Keywords: parabolic quotients
zbMATH Keywords: generalized flag manifolds
zbMATH Keywords: \(q=-1\) phenomenon
zbMATH Keywords: Weyl groups
zbMATH Keywords: finite irreducible Coxeter systems
MaRDI profile type: Publication
full work available at URL: s00026-002-8036-x
OpenAlex ID: W2014079140
DOI: 10.1007/S00026-002-8036-X
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description: scientific article; zbMATH DE number 1887494