Vertices of simple modules of symmetric groups labelled by hook partitions. (Q2260302)

This paper concerns modular representation theory of finite groups. Recall that if \(G\) is a group and \(M\) an indecomposable module for \(G\) in characteristic \(p\), then a \textit{vertex} of \(M\) is a subgroup \(P\) of \(G\) which is minimal subject to the condition that \(M\) appears as a direct summand of some module induced from \(P\) to \(G\). The vertices of \(M\) are all conjugate \(p\)-subgroups of \(G\), and provide a measure of the complexity of \(M\); for example, \(M\) is projective if and only the trivial group is a vertex of \(M\). Vertices are very difficult to calculate in general. This paper considers the case of the symmetric groups, computing vertices of the simple modules labelled (in James's notation) with the partitions \((n-r,1^r)\), where \(r<p\). In fact, this problem had already been solved by various authors except in one difficult case, where \(p>2\), \(r=p-1\) and \(n\equiv p\bmod p^2\). The present authors address this case, showing that (as expected) the vertices of these simple modules are the Sylow \(p\)-subgroups. As usual with these authors, the paper is extremely well written, striking a fine balance between economy and readability. The proof involves quite explicit calculations with bases of these simple modules, but the authors manage to make this as painless as possible. It is difficult to see how this work will generalise, but it is nonetheless an important contribution to the study of vertices.