Tensor products of direct sums (Q2368534)

The alternating \(n\)-fold tensor product \(\bigotimes_a^n E\) of a vector space \(E\) is the range of the antisymmetrization operator \(A: \bigotimes^n E \rightarrow \bigotimes^n E\) defined as \(A(x_1 \otimes \ldots \otimes x_n):= \frac{1}{n!} \sum_{\alpha \in S_n} \chi (\alpha) x_{\alpha(1)} \otimes \ldots x_{\alpha(n)}\) for elementary tensors (\(\chi(\alpha)\) denoting the sign of a permutation \(\alpha\)) and extended by linearity to the whole of \(\bigotimes^n E\). The authors prove as an analogue of a result in [\textit{J.~M.\ Ansemil} and \textit{K.~Floret}, Stud.\ Math.\ 129, 285--295 (1998; Zbl 0931.46005)] (for the symmetric tensor product) that if \(E=F_1 \oplus F_2\), then \(\bigotimes_a^n E\cong \bigoplus_{k=0}^n (\bigotimes_a^k F_1) \otimes (\bigotimes_a^{n-k} F_2)\). The proof is essentially different from the one for symmetric tensor products, one reason being that the latter is the linear span of the vectors \(\bigotimes^n x=\bigotimes_s^n x\), whereas \(\bigotimes_a^n x=0\) for all \(x\). A similar result for the \(3\)-fold Jacobian tensor product is also derived as well as topological variants for (stable) locally convex spaces. A~connection between these problems and irreducible group representations is made.