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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.08208 |
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| _version_ | 1866929592338481152 |
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| author | Evangelou, Charalampos |
| author_facet | Evangelou, Charalampos |
| contents | Let $G=GL_n(K)$ be the general linear group defined over an infinite field $K$ of positive characteristic $p$ and let $Δ(λ)$ be the Weyl module of $G$ which corresponds to a partition $λ$. In this paper we classify all homomorphisms $Δ(λ) \to Δ(μ)$ when $λ=(a,b,1^d)$ and $μ=(a+d,b)$, $d>1$. In particular, we show that $Hom_G(Δ(λ),Δ(μ))$ is nonzero if and only if $p=2$ and $a$ is even. In this case, we show that the dimension of the homomorphism space is equal to 1 and we provide an explicit generator whose description depends on binary expansions of various integers. We also show that these generators in general are not compositions of Carter-Payne homomorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08208 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On homomorphisms between Weyl modules: The case of a column transposition Evangelou, Charalampos Representation Theory Let $G=GL_n(K)$ be the general linear group defined over an infinite field $K$ of positive characteristic $p$ and let $Δ(λ)$ be the Weyl module of $G$ which corresponds to a partition $λ$. In this paper we classify all homomorphisms $Δ(λ) \to Δ(μ)$ when $λ=(a,b,1^d)$ and $μ=(a+d,b)$, $d>1$. In particular, we show that $Hom_G(Δ(λ),Δ(μ))$ is nonzero if and only if $p=2$ and $a$ is even. In this case, we show that the dimension of the homomorphism space is equal to 1 and we provide an explicit generator whose description depends on binary expansions of various integers. We also show that these generators in general are not compositions of Carter-Payne homomorphisms. |
| title | On homomorphisms between Weyl modules: The case of a column transposition |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2411.08208 |