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

TL;DR

This work analyzes the representation dimension of algebraic tori over arbitrary fields by translating the problem into the symmetric rank of $G$-lattices under a finite Galois action. It derives lower bounds on the maximal representation dimension $\operatorname{rdim}(n)$ via root systems and Weyl groups, and obtains exact values $\operatorname{rdim}_{\operatorname{irr}}(n)$ for many ranks by classifying irreducible maximal finite subgroups of $\operatorname{GL}_n(\mathbb{Z})$, aided by GAP data and a $\Theta$-series bound. The paper then develops prime-dimension theory, establishing asymptotic bounds (and conjectural extensions under Artin-type assumptions) that $\operatorname{rdim}_{\operatorname{irr}}(p)=2^p$ for many primes $p$ and bounding $\operatorname{symrank}(\mathbb{Z}^p, G) \le 2^p$ for almost-simple groups in prime dimensions. Collectively, these results illuminate the complexity of faithful torus representations across fields and connect to essential-dimension phenomena through $G$-lattice methods.

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 $n$.

Theorems & Definitions (82)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Definition 2.5