The asymptotics of tensor powers of Foulkes modules (Q6600698)

Let \(k\) be a field of positive characteristic and let \(m\ge 1\) be an integer. The symmetric group \(S_{2m}\) acts naturally on the collection of all set partitions of \(\{1, \dots, 2m\}\) into \(m\) sets, each of size two. Let \(H^{(2^m)}\) denote the corresponding permutation representation of \(S_{2m}\) over \(k\). In other words, \(H^{(2^m)}\) is the \(kS_{2m}\)-module induced from the trivial representation of the imprimitive wreath product \(S_2 \wr S_m \le S_{2m}\). The authors call \(H^{(2^m)}\) a Foulkes module. The goal of this paper is to study the asymptotic behavior of the non-projective part of tensor powers of these Foulkes modules. In particular, the authors determine the gamma invariant of these modules, as defined by \textit{D. Benson} and \textit{P. Symonds} [J. Lond. Math. Soc., II. Ser. 101, No. 2, 828--856 (2020; Zbl 1481.20027)].