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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2507.15505 |
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| _version_ | 1866913950985093120 |
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| author | Turek, Pavel Wang, Jialin |
| author_facet | Turek, Pavel Wang, Jialin |
| contents | Let $p$ be a prime and $n\geq 2$ be a positive integer. We establish new formulae for the decompositions of the first $p-1$ symmetric powers of the Specht module $S^{(n-1,1)}$ and the irreducible module $D^{(n-1,1)}$ in characteristic $p$ as direct sums of Young permutation modules. As an application of the formulae, we show that these symmetric powers have Specht filtration and find the vertices of their indecomposable summands. Our main tool, constructed in this paper, is a lift of a splitting map of a short exact sequence of certain symmetric powers to a splitting map of a short exact sequence of higher symmetric powers. This is a general construction, which can be applied to a broader family of modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetric powers of $S^{(n-1,1)}$ and $D^{(n-1,1)}$ Turek, Pavel Wang, Jialin Representation Theory 20C30 (Primary), 13A50, 20C20 (Secondary) Let $p$ be a prime and $n\geq 2$ be a positive integer. We establish new formulae for the decompositions of the first $p-1$ symmetric powers of the Specht module $S^{(n-1,1)}$ and the irreducible module $D^{(n-1,1)}$ in characteristic $p$ as direct sums of Young permutation modules. As an application of the formulae, we show that these symmetric powers have Specht filtration and find the vertices of their indecomposable summands. Our main tool, constructed in this paper, is a lift of a splitting map of a short exact sequence of certain symmetric powers to a splitting map of a short exact sequence of higher symmetric powers. This is a general construction, which can be applied to a broader family of modules. |
| title | Symmetric powers of $S^{(n-1,1)}$ and $D^{(n-1,1)}$ |
| topic | Representation Theory 20C30 (Primary), 13A50, 20C20 (Secondary) |
| url | https://arxiv.org/abs/2507.15505 |