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

TL;DR

This paper investigates the minimal faithful embedding dimensions of algebraic tori of fixed dimension over arbitrary fields, establishing bounds and exact values for certain dimensions using lattice group action theory.

Contribution

It introduces the concept of representation dimension for algebraic tori and determines the maximum for irreducible tori in specific dimensions, proposing conjectures for infinitely many primes.

Findings

Established lower bounds on maximal representation dimensions for all n.

Calculated exact maximum values for irreducible tori in specific dimensions.

Proposed conjectures for infinitely many prime dimensions.

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 $n$.