  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.
