## Date of Award

8-19-2024

## Document Type

Open Access Dissertation

## Department

Mathematics

## First Advisor

Alexander Duncan

## Abstract

Algebraic tori over a field k are special examples of affine group schemes over *k*, such as the multiplicative group of the field or the unit circle. Any algebraic torus can be embedded into the group of invertible *n* x *n* matrices with entries in *k* for some *n*, and the smallest such n is called the representation dimension of that torus. Representation dimensions of algebraic tori can be studied via symmetric ranks of *G*-lattices. A *G*-lattice *L* is a group isomorphic to the additive group Z^{n} for some *n*, along with an action of a group *G* on *L*, and the symmetric rank of a *G*-lattice is the minimal size of a *G*-stable generating set of *L*.<.p>

In this work, we are interested in finding the maximal representation dimension of all *n*-dimensional algebraic tori over all fields, denoted rdim(*n*). With this in mind, we also study the maximal symmetric rank of an irreducible *G*-lattice of rank *n*, which we write as symrank_{irr}(*n*). Ultimately, we find lower bounds for rdim(*n*) for all dimensions *n*, and we prove that these bounds are realized over all number fields and conjecture the exact value of rdim*(n)* in all dimensions. Furthermore, we find exact values of symrank_{irr}*(n)* for all *n* = 1, 2,..., 10, 11, 13, 17, 19, 23 and all primes *n* for which the multiplicative order of 2 modulo *n* is *n* - 1 or (*n*-1)/2. Along the way, we find formulas for the symmetric ranks of all irreducible *G*-lattices when *G* is the Weyl group of an irreducible root system; in particular, such *G*-lattices help us realize our bounds on rdim(*n)* and symrank_{irr}*(n)*.

## Rights

© 2024, Jason Bailey Heath

## Recommended Citation

Heath, J. B.(2024). *Representation Dimensions of Algebraic Tori and Symmetric Ranks of G-Lattices.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/7831