On a conjecture of Gow and Kleshchev concerning tensor products
Let \(\Sigma_n\) denote the symmetric group of degree \(n\) and let \(F\) denote a field of characteristic \(2\). It is well known that the irreducible representations of \(\Sigma_n\) over \(F\) are labelled by \(2\)-regular partitions of \(n\), that is, partitions of \(n\) into distinct parts. Given a \(2\)-regular partition \(\lambda\) of \(n\), let \(D^\lambda\) denote the irreducible \(F\Sigma_n\)-module labelled by \(\lambda\). The trivial irreducible \(F\Sigma_n\)-module is the representation \(D^{(n)}\) corresponding to the partition \(\lambda=(n)\) of \(n\). When \(\lambda\) and \(\mu\) are partitions of \(n\) different from the 1-part partition \((n)\), an interesting question is to decide when the tensor product \(D^\lambda\otimes D^\mu\) is irreducible. \textit{C. Bessenrodt} and \textit{A. Kleshchev} have shown [in Bull. Lond. Math. Soc. 32, No. 3, 292-296 (2000)] that such an irreducible tensor product can occur only if \(n\) is even and all the parts of at least one of the partitions \(\lambda\) and \(\mu\) have the same parity. On the other hand, \textit{R. Gow} and \textit{A. Kleshchev} conjectured [in J. Algebra 221, No. 1, 60-89 (1999; Zbl 0943.20013)] that an irreducible tensor product of the kind considered above occurs if and only if \(n=4l+2\) for some positive integer \(l\) and, with a suitable choice of the partitions, \(\lambda=(2l+2,2l)\) and \(\mu\) is the 2-part partition \((4l+1-2j,2j+1)\), where \(0\leq j<l\). The irreducible \(F\Sigma_n\)-module \(D^{(2l+2,2l)}\) is in this case the spin module of dimension \(2^{2l}\), obtained from the basic spin representation in characteristic \(0\) of the double cover of \(\Sigma_n\). In the paper under review, the authors prove one part of the conjecture of Gow and Kleshchev, namely, that the tensor product \(D^{(2l+2,2l)}\otimes D^{(4l+1-2j,2j+1)}\) is irreducible and isomorphic to \(D^\nu\), where \(\nu\) is the 4-part partition \((2l+1-j,2l-j,j+1,j)\), known as the double of the partition \((4l+1-2j,2j+1)\). They accomplish this by showing that the tensor product and the given \(D^\nu\) have the same dimension. Since it was shown by Gow and Kleshchev that \(D^\nu\) occurs as a constituent of the tensor product with multiplicity one, the irreduciblity theorem follows. As the authors remark, it is quite surprising that they can obtain enough information to find the dimensions of modules such as \(D^\nu\), since dimension formulae are not generally available for \(D^\mu\) whenever \(\mu\) has three or more parts, and even the dimension formula for \(D^\mu\) when \(\mu\) is a 2-part partition is not very explicit. We remark that there are no other irreducible tensor products of the form \(D^{(2l+2,2l)}\otimes D^\mu\), except when \(\mu=(n)\).
scientific article; zbMATH DE number 1471040
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