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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2205.01978 |
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| _version_ | 1866910602179379200 |
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| author | Lim, Kay Jin Wang, Jialin |
| author_facet | Lim, Kay Jin Wang, Jialin |
| contents | This paper focuses on the rank varieties for modules over a group algebra $\mathbb{F}E$ where $E$ is an elementary abelian $p$-group and $p$ is the characteristic of an algebraically closed field $\mathbb{F}$. In the first part, we give a sufficient condition for a Green vertex of an indecomposable module containing an elementary abelian $p$-group $E$ in terms of the rank variety of the module restricted to $E$. In the second part, given a homogeneous algebraic variety $V$ , we explore the problem on finding a small module with rank variety $V$ . In particular, we examine the simple module $D^{(kp-p+1,1^{p-1})}$ for the symmetric group $\mathfrak{S}_{kp}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_01978 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Small modules with interesting rank varieties Lim, Kay Jin Wang, Jialin Representation Theory 20C20(Primary) 20C30, 14J25 (Secondary) This paper focuses on the rank varieties for modules over a group algebra $\mathbb{F}E$ where $E$ is an elementary abelian $p$-group and $p$ is the characteristic of an algebraically closed field $\mathbb{F}$. In the first part, we give a sufficient condition for a Green vertex of an indecomposable module containing an elementary abelian $p$-group $E$ in terms of the rank variety of the module restricted to $E$. In the second part, given a homogeneous algebraic variety $V$ , we explore the problem on finding a small module with rank variety $V$ . In particular, we examine the simple module $D^{(kp-p+1,1^{p-1})}$ for the symmetric group $\mathfrak{S}_{kp}$. |
| title | Small modules with interesting rank varieties |
| topic | Representation Theory 20C20(Primary) 20C30, 14J25 (Secondary) |
| url | https://arxiv.org/abs/2205.01978 |