On stability and nonvanishing of homomorphism spaces between Weyl modules

Consider the general linear group $G=G{L}_n(K)$ defined over an infinite field $K$ of positive characteristic $p$. We denote by $\mathrm{\Delta }(\lambda )$ 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 $Ho{m}_G(\mathrm{\Delta }(\lambda ),\mathrm{\Delta }(\mu ))$ $\simeq$ $Ho{m}_G(\mathrm{\Delta }(\lambda +\gamma ),\mathrm{\Delta }(\mu +\gamma ))$, 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 $Ho{m}_G(\mathrm{\Delta }(\lambda ),\mathrm{\Delta }(\mu ))$ is nonzero. An explicit map is provided that corresponds to the sum of all semistandard tableaux of shape $\mu$ and weight $\lambda$.