A remark on integral representations of \(GL_{{\mathbb{Z}}}(n)\) (Q1113286)

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.