On the rank varieties of some simple modules for symmetric groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910719804440576 |
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| author | Wang, Jialin |
| author_facet | Wang, Jialin |
| contents | In the previous work, Lim and the author determined the rank variety of the simple $\mathbb{F}\mathfrak{S}_{kp}$-module $D(p-1)=D^{(kp-p+1,1^{p-1})}$ with respect to some maximal elementary abelian $p$-subgroup $E_k$ and the complexity when $k\not\equiv 1\pmod p$ and $p$ is odd. Their method relied on the dimension of the module, which is dependent on $k$. In this paper, we extend this result to the case for any $k\geq 2$, and determine the rank variety of $D(p-1)$ and its complexity, providing a proof independent of $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19243 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the rank varieties of some simple modules for symmetric groups Wang, Jialin Representation Theory 20C20, 20C30 In the previous work, Lim and the author determined the rank variety of the simple $\mathbb{F}\mathfrak{S}_{kp}$-module $D(p-1)=D^{(kp-p+1,1^{p-1})}$ with respect to some maximal elementary abelian $p$-subgroup $E_k$ and the complexity when $k\not\equiv 1\pmod p$ and $p$ is odd. Their method relied on the dimension of the module, which is dependent on $k$. In this paper, we extend this result to the case for any $k\geq 2$, and determine the rank variety of $D(p-1)$ and its complexity, providing a proof independent of $k$. |
| title | On the rank varieties of some simple modules for symmetric groups |
| topic | Representation Theory 20C20, 20C30 |
| url | https://arxiv.org/abs/2411.19243 |