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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2502.15513 |
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| _version_ | 1866915165408067584 |
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| author | Heath, Bailey |
| author_facet | Heath, Bailey |
| contents | We define the representation dimension of an algebraic torus $T$ to be the minimal positive integer $r$ such that there exists a faithful embedding $T \hookrightarrow \operatorname{GL}_r$. Given a positive integer $n$, there exists a maximal representation dimension of all $n$-dimensional algebraic tori over all fields. In this paper, we use the theory of group actions on lattices to find lower bounds on this maximum for all $n$. Further, we find the exact maximum value for irreducible tori for all $n \in \left\lbrace 1, 2, \dots, 10, 11, 13, 17, 19, 23\right\rbrace$ and conjecturally infinitely many primes $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal Representation Dimensions of Algebraic Tori of Fixed Dimension Over Arbitrary Fields Heath, Bailey Algebraic Geometry Group Theory 20G15 (Primary), 20C10 (Secondary) We define the representation dimension of an algebraic torus $T$ to be the minimal positive integer $r$ such that there exists a faithful embedding $T \hookrightarrow \operatorname{GL}_r$. Given a positive integer $n$, there exists a maximal representation dimension of all $n$-dimensional algebraic tori over all fields. In this paper, we use the theory of group actions on lattices to find lower bounds on this maximum for all $n$. Further, we find the exact maximum value for irreducible tori for all $n \in \left\lbrace 1, 2, \dots, 10, 11, 13, 17, 19, 23\right\rbrace$ and conjecturally infinitely many primes $n$. |
| title | Maximal Representation Dimensions of Algebraic Tori of Fixed Dimension Over Arbitrary Fields |
| topic | Algebraic Geometry Group Theory 20G15 (Primary), 20C10 (Secondary) |
| url | https://arxiv.org/abs/2502.15513 |