Finite reflection groups and linear preserver problems
type: Item
instance of: scholarly article
title: Finite reflection groups and linear preserver problems
zbMATH Open document ID: 1060.15007
author: Chi-Kwong Li
author: Nahum Zobin
author: Ilya M. Spitkovsky
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-09-27T00:00:00Z
review text: Let \(G\) be a Coxeter group and \(V\) a Euclidean space such that \(G\subset \text{End}\,V\). Let \({\mathcal L}(G)\) be the set of linear transformations \(\Phi\) in \(\text{ End}\,V\) such that \(\Phi(G)=G\). Let \(P,Q\) be in the normalizer \(N(G)\) of \(G\) in the orthogonal group \(O(V)\) and assume that \(PQ\in G\). Let \({\mathcal RE}(G)\) be the set of transformations \(X\to PXQ\) and \(X\to PX^*Q\), where \(X^*\) is the adjoint of \(X\). Then \({\mathcal RE}(G)\subset{\mathcal L}(G)\). The authors give a complete description of the set \({\mathcal L}(G)\) for each of the four infinite families of finite irreducible Coxeter groups, i.e., of \(A_n\), \(B_n\), \(D_n\), and \(I_2(n)\). They show that \({\mathcal RE}(G)={\mathcal L}(G)\) except for \(G=B_n\).
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 2103626
zbMATH Keywords: linear preserver problem
zbMATH Keywords: finite reflection group
zbMATH Keywords: Euclidean space
zbMATH Keywords: Coxeter group
zbMATH Keywords: linear transformations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069902
OpenAlex ID: W1979531272
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DOI: 10.1216/RMJM/1181069902
description: scientific article; zbMATH DE number 2103626