Relating homomorphism spaces between Specht modules of different degrees
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866916239543107584 |
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| author | Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia |
| author_facet | Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia |
| contents | Let $K$ be an infinite field of characteristic $p>0$ and let $λ, μ$ be partitions of $n$, where $λ=(λ_1,...,λ_n)$ and $μ=(μ_1,..,μ_n)$. By $S^λ$ we denote the Specht module corresponding to $λ$ for the group algebra $K\mathfrak{S}_n$ of the symmetric group $\mathfrak{S}_n$. D. Hemmer has raised the question of relating the homomorphism spaces $\Hom_{\mathfrak{S}_n}(S^μ, S^λ)$ and $\Hom_{\mathfrak{S}_{n'}}(S^{μ^+}, S^{λ^+})$, where $n'=n+kp^d$, $λ^+ =λ+(kp^{d})$, $μ^+=μ+(kp^{d})$, and $d, k$ are positive integers. We show that these are isomorphic if $p$ is odd, $p^d >\min\{λ_2, μ_1-λ_1\}$ and $μ_2 \le λ_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_05733 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Relating homomorphism spaces between Specht modules of different degrees Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia Representation Theory Let $K$ be an infinite field of characteristic $p>0$ and let $λ, μ$ be partitions of $n$, where $λ=(λ_1,...,λ_n)$ and $μ=(μ_1,..,μ_n)$. By $S^λ$ we denote the Specht module corresponding to $λ$ for the group algebra $K\mathfrak{S}_n$ of the symmetric group $\mathfrak{S}_n$. D. Hemmer has raised the question of relating the homomorphism spaces $\Hom_{\mathfrak{S}_n}(S^μ, S^λ)$ and $\Hom_{\mathfrak{S}_{n'}}(S^{μ^+}, S^{λ^+})$, where $n'=n+kp^d$, $λ^+ =λ+(kp^{d})$, $μ^+=μ+(kp^{d})$, and $d, k$ are positive integers. We show that these are isomorphic if $p$ is odd, $p^d >\min\{λ_2, μ_1-λ_1\}$ and $μ_2 \le λ_1$. |
| title | Relating homomorphism spaces between Specht modules of different degrees |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2108.05733 |