Large dimension homomorphism spaces between Specht modules for symmetric groups

Let F be a field of characteristic p. We show that Hom_FΣn(S^λ,S^μ) can have arbitrarily large dimension as n and p grow, where S^λ and S^μ are Specht modules for the symmetric group Σ_n. Similar results hold for the Weyl modules of the general linear group. Every previously computed example has been at most one-dimensional, with the exception of Specht modules over a field of characteristic two. The proof uses the work of Chuang and Tan, providing detailed information about the radical series of Weyl modules in Rouquier blocks.