On minimal lengths of expressions of Coxeter group elements as products of reflections
type: Item
instance of: scholarly article
title: On minimal lengths of expressions of Coxeter group elements as products of reflections
zbMATH Open document ID: 0980.20028
DOI: 10.1090/S0002-9939-01-05876-2
published in: Proceedings of the American Mathematical Society
publication date: 2001-06-06T00:00:00Z
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1602158
zbMATH Keywords: Coxeter groups
zbMATH Keywords: lengths of words
zbMATH Keywords: products of reflections
zbMATH Keywords: Bruhat graphs
zbMATH Keywords: \(R\)-polynomials
zbMATH Keywords: simple reflections
author: Matthew Dyer
MaRDI profile type: Publication
cites work: Q4320804
cites work: A combinatorial formula for Kazhdan-Lusztig polynomials
cites work: Q5668957
cites work: On some geometric aspects of Bruhat orderings. I: A finer decomposition of Bruhat cells
cites work: Reflection subgroups of Coxeter systems
cites work: Q3977655
cites work: Q4029511
cites work: Q4285662
cites work: Q3998721
cites work: Representations of Coxeter groups and Hecke algebras
cites work: Invariants of Finite Reflection Groups
full work available at URL: s0002-9939-01-05876-2
OpenAlex ID: W1510366593
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review text: Let \(W\) be a Coxeter group and let \(T\) be the set of all reflections (not only simple reflections) in \(W\). Let \(w\in W\). The length \(\ell'(w)\) of \(w\) is the minimal number of factors occurring amongst all expressions of \(w\) as a product of elements in \(T\). If \(W\) is finite, then \(\ell'(w)\) is the codimension of the \(1\)-eigenspace of \(w\).NEWLINENEWLINENEWLINEThe author shows that \(\ell'(w)\) is equal to the minimum number of simple reflections that must be deleted from a fixed reduced expression of \(w\) so that the resulting product of simple reflections is equal to the identity element of \(W\). He also characterizes the length \(\ell'(w)\) in terms of combinatorial data related to the Bruhat graph and to the \(R\)-polynomial \(R_{e,w}\).
reviewed by: Erich W. Ellers
description: scientific article; zbMATH DE number 1602158