Submodule structure of some Specht modules

Abstract

In this paper we show how homomorphisms between certain permutation modules for the symmetric group S, may be used to give information about the submodule structure of Specht modules for hook and two-rowed diagrams. We continue the investigation of the properties of the hook module at characteristic 2 which was begun in [3,4]. Here we give precise conditions for the module to be completely reducible, and to have a unique minimal submodule. We also show that the two-rowed Specht module has a unique minimal submodule over a field of any characteristic. Let K be any field, and X = (x i ,..., xn} a set of independent commutative indeterminates. We may define the action of S, on any polynomal in the elements of X by u(xI) = xu(,) for each u in S,. We denote by SCnPrVr) the Specht module corresponding to a Young diagram with n r nodes in the first row and r nodes in the second, and by S(n-rV1” the module for a diagram with n r nodes in the top row and r rows of length 1. Then S(n-rlr’ and S(n-r*“) are generated over KS,, by T, and U,, respectively, where