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A remark on integral representations of \(GL_{{\mathbb{Z}}}(n)\). (English) Zbl 0662.20034

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

MSC:

20G05 Representation theory for linear algebraic groups
20C12 Integral representations of infinite groups
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References:

[1] Buchsbaum, D.; Akin, Characteristic-free representation theory of the general linear group, Adv. in Math., 58, 149-200 (1985) · Zbl 0607.20021
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