Bases for representation rings of Lie groups and their maximal tori

TL;DR

This paper explicitly constructs a basis for the representation ring of the maximal torus of SU(n) within the representation ring of the Lie group, extending previous theoretical results with concrete examples.

Contribution

It provides an explicit basis for the representation ring of the maximal torus of SU(n), complementing prior theoretical work by Pittie and Steinberg.

Findings

Explicit basis for R(T) of SU(n) derived

Computations performed for SO(2n)

Enhanced understanding of representation rings in Lie groups

Abstract

A Lie group is a group that is also a differentiable manifold, such that the group operation is continuous respect to the topological structure. To every Lie group we can associate its tangent space in the identity point as a vector space, which is its Lie algebra. Killing and Cartan completely classified simple Lie groups into seven types. Representation of a Lie group is a homomorphism from the Lie group to the automorphism group of a vector space. In general represenations of Lie group are determined by its Lie algebra and its the connected components. We may consider operations like direct sum and tensor product with respect to which the representations of G form a ring R(G). Assume the group is compact. For every Lie group we may find a maximal torus inside of it. By projecting the representation ring of the Lie group to the representation ring of its maximal torus, we may consider to express R(T) elements as sums of products of base elements with R(G) elements. In 1970s Pittie and Steinberg proved R(T) is a free module over R(G) using homological algebra methods without specifying the bases in each seven types. In this senior project I found an explicit basis for the representation ring of the maximal torus of Lie group SUn in terms of the represenation ring of the Lie group itself. I also did some computation with SO2n.