A Gelfand model for Weyl groups of type \(D_{2n}\).
type: Item
instance of: scholarly article
title: A Gelfand model for Weyl groups of type \(D_{2n}\).
zbMATH Open document ID: 1250.20004
DOI: 10.5402/2012/658201
author: José O. Araujo
author: Luis C. Maiarú
author: Mauro Natale
published in: ISRN Algebra
publication date: 2012-09-07T00:00:00Z
review text: Summary: A Gelfand model for a finite group \(G\) is a complex representation of \(G\), which is isomorphic to the direct sum of all irreducible representations of \(G\). When \(G\) is isomorphic to a subgroup of \(\text{GL}_n(\mathbb C)\), where \(\mathbb C\) is the field of complex numbers, it has been proved that each \(G\)-module over \(\mathbb C\) is isomorphic to a \(G\)-submodule in the polynomial ring \(\mathbb C[x_1,\dots,x_n]\), and taking the space of zeros of certain \(G\)-invariant operators in the Weyl algebra, a finite-dimensional \(G\)-space \(\mathcal N_G\) in \(\mathbb C[x_1,\dots,x_n]\) can be obtained, which contains all the simple \(G\)-modules over \(\mathbb C\). This type of representation has been named polynomial model. It has been proved that when \(G\) is a Coxeter group, the polynomial model is a Gelfand model for \(G\) if, and only if, \(G\) has not an irreducible factor of type \(D_{2n}\), \(E_7\), or \(E_8\). This paper presents a model of Gelfand for a Weyl group of type \(D_{2n}\) whose construction is based on the same principles as the polynomial model.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 6078649
zbMATH Keywords: Gel'fand models
zbMATH Keywords: finite groups
zbMATH Keywords: complex representations
zbMATH Keywords: irreducible representations
zbMATH Keywords: Weyl groups
zbMATH Keywords: Coxeter groups
zbMATH Keywords: polynomial models
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.5402/2012/658201
OpenAlex ID: W2153963423
Wikidata QID: Q58691989
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description: scientific article; zbMATH DE number 6078649