Representation Dimensions of Algebraic Tori and Symmetric Ranks of G-Lattices

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 Zn 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 symrankirr(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 symrankirr(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 symrankirr(n).