Large dimension homomorphism spaces between Specht modules for symmetric groups

Let F be a field of characteristic p. We show that HomFΣn(Sλ,Sμ) can have arbitrarily large dimension as n and p grow, where Sλ and Sμ are Specht modules for the symmetric group Σn. 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.