On the socle of Weyl modules (Q749666)

For a field k of characteristic p (p prime or 0) and to each partition \(\lambda =(\lambda_ 1,...,\lambda_ n)\), \(\lambda_ 1\geq \lambda_ 2\geq...\geq \lambda_ n\geq 0\), of a natural number \(n\geq 1\) \textit{R. W. Carter} and \textit{G. Lusztig} constructed a module \(V_{\lambda,k}\), which they called Weyl module [Math. Z. 136, 193-242 (1974; Zbl 0298.20009)]. These modules have the important property that they have a unique simple factor module \(F_{\lambda,k}\). Fixing n, these modules give a full set of simple modules for the Schur algebra \(S_ k(n,n)\). In case k has infinitely many elements these results can be interpreted for the polynomial representation theory of the \(GL_ n(k)\). Regarded as modules for the Schur algebra no such restriction is necessary. \textit{S. E. Fettes} considered in [Ph.D. thesis, Univ. Mass. (1982)] the socle of Weyl modules for some particular partitions, all of which happen to be simple modules. We extend her ideas to derive a relation between the socle of \(V_{\lambda,k}\) and the socle of the dual \(S_ k^{\lambda \#}\) of the Specht modules \(S_ k^{\lambda}\).