Kempf, George R. A remark on integral representations of \(GL_{{\mathbb{Z}}}(n)\). (English) Zbl 0662.20034 J. Algebra 115, No. 2, 340-341 (1988). 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. Reviewer: J.H.Lindsey II Cited in 1 Document MSC: 20G05 Representation theory for linear algebraic groups 20C12 Integral representations of infinite groups Keywords:divided powers; general linear group; exterior power; symmetric power; integral representation; diagonal representation PDFBibTeX XMLCite \textit{G. R. Kempf}, J. Algebra 115, No. 2, 340--341 (1988; Zbl 0662.20034) Full Text: DOI References: [1] Buchsbaum, D.; Akin, Characteristic-free representation theory of the general linear group, Adv. in Math., 58, 149-200 (1985) · Zbl 0607.20021 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.