\(p\)-restriction of partitions and homomorphisms between Specht modules. (Q855988)

Let \(\lambda\) be a partition of \(n\) and \(\mathbb{S}_n\) be the symmetric group on \(n\) letters. Let \(F\) be a field of characteristic \(p\). Let \(S^\lambda\) denote a Specht module for the group algebra \(F\mathbb{S}_n\). An interesting problem concerning Specht modules is the determination of the homomorphism space \(\Hom_{F\mathbb{S}_n}(S^\lambda,S^\mu)\) for partitions \(\lambda\) and \(\mu\) of \(\mathbb{S}_n\). The authors, using elementary methods of Young tableaux, prove that \(\Hom_{F\mathbb{S}_n}(S^\lambda,S^\mu)\) is one-dimensional whenever \(\mu\) is the \(p\)-restriction of \(\lambda\).