Large Dimension Homomorphism Spaces Between Specht Modules for Symmetric Groups

Let $F$ be a field of characteristic $p$. We show that $\Hom_{FΣ_n}(S^λ, S^μ)$ 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.

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