On homomorphisms involving a hook Weyl module

TL;DR

This paper characterizes all homomorphisms between Weyl modules for $GL_n(k)$ when one partition is a hook, and explores implications for algebraic groups of types B, C, and D.

Contribution

It provides a complete description of homomorphisms involving hook Weyl modules and extends results to other classical algebraic groups.

Findings

All homomorphisms between Weyl modules with a hook partition are determined.

Nonvanishing homomorphisms exist between Weyl modules for types B, C, D under certain hook conditions.

The results unify understanding of Weyl module homomorphisms across different group types.

Abstract

Let $k$ be an infinite field of positive characteristic. We determine all homomorphisms between Weyl modules for $GLn(k)$, where one of the partitions is a hook. As a consequence we obtain a nonvanishing result concerning homomorphisms between Weyl modules for algebraic groups of type B, C and D when one of the partitions is a hook.