Branching rules for modular representations of symmetric groups. IV
[For part III see J. Lond. Math. Soc., II. Ser. 54, No. 1, 25-38 (1996; Zbl 0854.20014).] Let \(\Sigma_n\) be the symmetric group on \(n\) letters, and \(D\) be an irreducible representation of \(\Sigma_n\) over a ground field \(F\) of characteristic \(p>0\). The main results of this paper are: (1) We describe the algebra \(\text{End}_{\Sigma_{n-1}}(D \downarrow_{\Sigma_{n-1}})\); (2) we obtain a criterion for the semisimplicity of the induced module \(D \uparrow^{\Sigma_{n+1}}\), and in the case of semisimplicity describe its composition factors. We note that for characteristic \(p=2\) the criterion for the semisimplicity we get was conjectured by Benson. Thus we generalize to an arbitrary \(p\) and prove the last unproved conjecture made by Benson at the Arcata Conference. Actually, (2) easily follows from (1) using a Mackey type theorem together with our previous results on \(D \uparrow^{\Sigma_{n+1}}\).
scientific article; zbMATH DE number 1214114