On stability and nonvanishing of homomorphism spaces between Weyl modules

TL;DR

The paper addresses the problem of stability and nonvanishing of Hom spaces between Weyl modules Δ(λ) for G = GL_n(K) in positive characteristic. It develops a combinatorial framework based on the ABW presentation, weight-space decomposition, and semistandard tableaux to derive two main results: (i) a stability/periodicity theorem showing dim Hom_G(Δ(λ),Δ(μ)) = dim Hom_G(Δ(λ+γ),Δ(μ+γ)) under explicit p-adic divisibility conditions on γ, and corollaries about stabilization of dimensions along p-powers and periodicity for suitable ν; and (ii) a nonvanishing criterion with an explicit map Δ(λ) → Δ(μ) given by the sum over SST_λ(μ), under a different set of combinatorial hypotheses. These results extend Hemmer’s question on homomorphisms, connect to Carter-Payne phenomena, and provide concrete tools for understanding Weyl-module homomorphisms via tableau combinatorics and p-adic analysis.

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 $λ$.

Theorems & Definitions (50)

• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• Definition 2.4
• Remark 2.5
• Example
• Proposition 2.6: AB
• Lemma 2.7
• proof
• Corollary 2.8