Two theorems on the vertices of Specht modules. (Q1423824)

Let \(F\) be a field of characteristic \(p>0\), and let \(G\) denote the symmetric group \(S_n\) of degree \(n\). The author proves two results on vertices of indecomposable Specht modules for \(FG\): 1. An indecomposable Specht module \(S^\lambda\) has a non-trivial cyclic vertex if and only if \(\lambda\) has \(p\)-weight one. 2. If \(p\) does not divide \(n\) and \(S^{(n-r,1^r)}\) is indecomposable then its vertex is a Sylow \(p\)-subgroup of \(S_{n-r-1}\times S_r\).