On stability and nonvanishing of homomorphism spaces between Weyl modules
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arXiv
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| Format: | Preprint |
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2023
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| author | Evangelou, Charalambos Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia |
| author_facet | Evangelou, Charalambos Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia |
| contents | 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 $λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_14203 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On stability and nonvanishing of homomorphism spaces between Weyl modules Evangelou, Charalambos Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia Representation Theory 20G05, 05E10 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 $λ$. |
| title | On stability and nonvanishing of homomorphism spaces between Weyl modules |
| topic | Representation Theory 20G05, 05E10 |
| url | https://arxiv.org/abs/2310.14203 |