Carter-Payne homomorphisms and branching rules for endomorphism rings of Specht modules

Let Sigma(n) be the symmetric group of degree n, and let F be a field of characteristic p. Suppose that l is a partition of n + 1, that alpha and eta are partitions of n that can be obtained by removing a node of the same residue from lambda, and that alpha dominates beta. Let S(alpha) and S(beta) be the Specht modules, defined over F, corresponding to alpha, respectively beta. We use Jucys-Murphy elements to give a very simple description of a non-zero homomorphism S(alpha) -> S(beta). Following Lyle, we also give an explicit expression for the homomorphism in terms of semi-standard homomorphisms. Our methods furnish a lower bound for the Jantzen submodule of S(beta) that contains the image of the homomorphism. Our results allow us to describe completely the structure of the ring End(F Sigma n) (S(lambda) down arrow Sigma(n)) when p not equal 2.