On homomorphisms into Weyl modules corresponding to partitions with two parts

TL;DR

This paper establishes arithmetic conditions under which nonzero homomorphisms exist between Weyl modules for $GL_n(K)$ when the target partition has two parts, revealing new insights into their structure in positive characteristic.

Contribution

It provides new sufficient conditions for the existence and dimension of homomorphism spaces between Weyl modules with specific partition shapes in positive characteristic.

Findings

Identifies arithmetic conditions for nonzero homomorphisms

Determines when homomorphism spaces have dimension at least 2

Focuses on partitions with two parts in Weyl modules

Abstract

Let $K$ be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, \mu$ for the existence of a nonzero homomorphism $\Delta(\lambda) \to \Delta (\mu)$ of Weyl modules for the general linear group $GL_n(K)$. Also for each $p$ we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.