On stability and nonvanishing of homomorphism spaces between Weyl modules

Abstract

Consider the general linear group $G=G{L}_n(K)$ defined over an infinite field $K$ of positive characteristic $p$. We denote by $\Delta (\lambda )$ the Weyl module of $G$ which corresponds to a partition $\lambda$. Let $\lambda ,\mu$ be partitions of $r$ and let $\gamma$ be partition with all parts divisible by $p$. In the first main result of this paper, we find sufficient conditions on $\lambda ,\mu$ and $\gamma$ so that $Ho{m}_G(\Delta (\lambda ),\Delta (\mu ))$ $\simeq$ $Ho{m}_G(\Delta (\lambda +\gamma ),\Delta (\mu +\gamma ))$, thus providing an answer to a question of D. Hemmer. As corollaries we obtain stability and periodicity results for homomorphism spaces. In the second main result we find related sufficient conditions on $\lambda ,\mu$ and $p$ so that $Ho{m}_G(\Delta (\lambda ),\Delta (\mu ))$ is nonzero. An explicit map is provided that corresponds to the sum of all semistandard tableaux of shape $\mu$ and weight $\lambda$.

The referee's comments and suggestions have been incorporated. Accepted for publication in Algebraic Combinatorics

References in corpus (2)

• On homomorphisms into Weyl modules corresponding to partitions with two parts
• A periodicity theorem for extensions of Weyl modules