The image of the Specht module under the inverse Schur functor in arbitrary characteristic

On the Endomorphism Algebra of Specht Modules in Even Characteristic

“…This functor is a right-inverse of f and it is right-exact. For α ∈ Λ(n, r) one has that gM (α) ∼ = S α E [DG 2 , Appendix A], and for λ ∈ Λ + (n, r) and p = 2 one has that gSp(λ) ∼ = ∇(λ) [D 3 , Theorem 10.6 (i)], [McD,Theorem 1.1]. By Frobenius reciprocity, it follows that for V ∈ M k (n, r) and W ∈ kS r -mod, there is a k-isomorphism Hom G (gW, V ) ∼ = Hom kSr (W, f V ).…”