Large Dimension Homomorphism Spaces Between Specht Modules for Symmetric Groups

TL;DR

This paper demonstrates that the homomorphism spaces between Specht modules for symmetric groups can grow arbitrarily large in dimension as the group size and characteristic increase, revealing new complexity in modular representation theory.

Contribution

It establishes that the dimensions of homomorphism spaces between Specht modules are unbounded, extending understanding beyond previously known small-dimensional cases.

Findings

Hom spaces can have arbitrarily large dimension as n and p grow

Previous examples of large hom spaces were limited to characteristic two

Detailed analysis of Weyl modules' radical series supports the results

Abstract

Let $F$ be a field of characteristic $p$. We show that $\Hom_{F\Sigma_n}(S^\lambda, S^\mu)$ can have arbitrarily large dimension as $n$ and $p$ grow, where $S^\lambda$ and $S^\mu$ are Specht modules for the symmetric group $\Sigma_n$. 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.