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arXiv · 2502.15513

Maximal Representation Dimensions of Algebraic Tori of Fixed Dimension Over Arbitrary Fields

Bailey Heath
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Abstract

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 $n$.

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Bailey Heath. 2025-02-21. Maximal Representation Dimensions of Algebraic Tori of Fixed Dimension Over Arbitrary Fields. https://arxiv.org/abs/2502.15513

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