A remark on integral representations of \(GL_{{\mathbb{Z}}}(n)\)
type: Item
instance of: scholarly article
title: A remark on integral representations of \(GL_{{\mathbb{Z}}}(n)\)
zbMATH Open document ID: 0662.20034
DOI: 10.1016/0021-8693(88)90261-X
author: George R. Kempf
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Consider the obvious representation of \(G=GL_ 2(n)\), the general linear group with integral coefficients on \(V=Z^{\otimes n}\). Let \(\bigwedge^ iV\) be the ith exterior power, \(S^ iV\) the ith symmetric power, and \(\Gamma^ iV\) the dual of the ith symmetric power of the dual of V. The author shows that any integral representation W is the quotient of direct sums of the diagonal representation of G on \(\Gamma^{i_ 1}V\otimes...\otimes \Gamma^{i_ n}V\otimes (\bigwedge^ nV)^{\otimes j}\). He also shows that the result fails for \(W=\Gamma^ 2V\) for \(n=2\) if the \(\Gamma^{i_ k}V\) are all replaced by V. A Hopf algebra is involved in the proof.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C12
zbMATH DE Number: 4081835
zbMATH Keywords: divided powers
zbMATH Keywords: general linear group
zbMATH Keywords: exterior power
zbMATH Keywords: symmetric power
zbMATH Keywords: integral representation
zbMATH Keywords: diagonal representation
MaRDI profile type: Publication
cites work: Characteristic-free representation theory of the general linear group
full work available at URL: 0021-8693(88)90261-x
OpenAlex ID: W2041479783
description: scientific article; zbMATH DE number 4081835