A peculiar property of exceptional Weyl groups
type: Item
instance of: scholarly article
title: A peculiar property of exceptional Weyl groups
zbMATH Open document ID: 0943.20014
DOI: 10.1023/A:1009946123738
author: Amy E. Ksir
published in: Algebras and Representation Theory
publication date: 2000-09-03T00:00:00Z
review text: Let \(W\) be an irreducible Weyl group and \(P\) be a parabolic subgroup of \(W\). Consider the irreducible components of \(\text{Ind}^W_P\mathbf{1}_P\), the induced representation of the trivial representation \(\mathbf{1}_P\) of \(P\) to \(W\). It is known that the trivial representation \(\mathbf{1}_W\) and the reflection representation \(\Lambda\) always occur as irreducible components of \(\text{Ind}^W_P\mathbf{1}_P\) for any proper parabolic subgroup \(P\) of \(W\), and that these are the only two that appear for \(W\) of classical types [see \textit{R. Donagi}, Astérisque 218, 145-175 (1993; Zbl 0820.14031)]. In the paper under review, the author shows that when \(W\) is of exceptional type, there is at least one irreducible representation \(\rho\) of \(W\), besides the trivial and reflection representation, such that \(\rho\) is a component of \(\text{Ind}^W_P\mathbf{1}_P\) for all proper parabolic subgroups \(P\) of \(W\). More precisely, when \(W\) is of exceptional type other than \(E_8\), such a \(\rho\) is unique, which is the nontrivial irreducible representation in the symmetric square of \(\Lambda\); when \(W\) is of type \(E_8\), \(\rho\) can be one of the three, which are in the symmetric square, symmetric cube and the symmetric quartic of \(\Lambda\), respectively.
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20C10
zbMATH DE Number: 1385251
zbMATH Keywords: irreducible Weyl groups
zbMATH Keywords: irreducible components
zbMATH Keywords: induced representations
zbMATH Keywords: reflection representations
zbMATH Keywords: irreducible representations
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: exceptional Weyl groups
MaRDI profile type: Publication
full work available at URL: a:1009946123738
OpenAlex ID: W2254384372
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description: scientific article; zbMATH DE number 1385251