Completely splittable representations of symmetric groups
This paper is motivated by earlier papers by \textit{A. Ryba} [J. Algebra 170, No. 2, 678-686 (1994; Zbl 0830.20026)] and \textit{O. Mathieu} [Lett. Math. Phys. 38, No. 1, 23-32 (1996; Zbl 0855.20010)]. Both of these papers concern the irreducible \(KS_n\)-modules \(D^\lambda\) corresponding to the \(p\)-regular partitions \(\lambda\) of \(n\), where \(K\) is a field of characteristic \(p>0\). The first paper concerns a certain family of \(KS_n\) in the case \(p=5\) which have a remarkable property on restriction to Young subgroups. The present author uses this to motivate his definition of \(D^\lambda\) being completely splittable. Furthermore, he gives a description of completely splittable representations and proves that \(D^\lambda\) is completely splittable if and only if \(\chi(\lambda)\) (the length of a certain hook in \(\lambda\)) \(\leq p\). Mathieu, in his paper, had proved a formula for the degree of such \(D^\lambda\) in terms of paths from (1) to \(\lambda\) in the modular Young graph. The present paper gives a new proof of this result which is based on ideas from an earlier paper [the author, J. Lond. Math. Soc., II. Ser. 54, No. 1, 25-38 (1996; Zbl 0854.20014)].
scientific article; zbMATH DE number 883775