On homomorphisms between Weyl modules: The case of a column transposition

TL;DR

This paper classifies homomorphisms between specific Weyl modules for the general linear group over fields of positive characteristic, revealing conditions under which these homomorphisms are nonzero and explicitly describing their generators.

Contribution

It provides a complete classification of homomorphisms between Weyl modules for a particular partition case, including explicit generators and their dependence on binary expansions.

Findings

Homomorphisms are nonzero only if p=2 and a is even.

Dimension of the homomorphism space is 1 in the nonzero case.

Generators are not generally compositions of Carter-Payne homomorphisms.

Abstract

Let $G=GL_n(K)$ be the general linear group defined over an infinite field $K$ of positive characteristic $p$ and let $\Delta(\lambda)$ be the Weyl module of $G$ which corresponds to a partition $\lambda$. In this paper we classify all homomorphisms $\Delta(\lambda) \to \Delta(\mu)$ when $\lambda=(a,b,1^d)$ and $\mu=(a+d,b)$, $d>1$. In particular, we show that $Hom_G(\Delta(\lambda),\Delta(\mu))$ is nonzero if and only if $p=2$ and $a$ is even. In this case, we show that the dimension of the homomorphism space is equal to 1 and we provide an explicit generator whose description depends on binary expansions of various integers. We also show that these generators in general are not compositions of Carter-Payne homomorphisms.