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

We define the representation dimension of an algebraic torus $T$ to be the minimal positive integer $r$ such that there exists a faithful embedding $T\hookrightarrow {\mathrm{GL}}_r$. 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 $n\in \{1,2,\dots ,10,11,13,17,19,23\}$ and conjecturally infinitely many primes $n$.