On stability and nonvanishing of homomorphism spaces between Weyl modules

Abstract

Consider the general linear group $G=GL_{n}(K)$ defined over an infinite field $K$ of positive characteristic $p$. We denote by $Δ(λ)$ the Weyl module of $G$ which corresponds to a partition $λ$. Let $λ, μ$ be partitions of $r$ and let $γ$ be partition with all parts divisible by $p$. In the first main result of this paper, we find sufficient conditions on $λ, μ$ and $γ$ so that $Hom_G(Δ(λ),Δ(μ))$ $ \simeq$ $ Hom_G(Δ(λ+γ),Δ(μ+γ))$, 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 $λ, μ$ and $p$ so that $Hom_G(Δ(λ),Δ(μ))$ is nonzero. An explicit map is provided that corresponds to the sum of all semistandard tableaux of shape $μ$ and weight $λ$.