In this project, we will first examine the representation theory of the unitary group and the combinatorial aspect of the relevant symmetric polynomials. Based on these we will further study the representation theory of other classical Lie groups and the smallest exceptional Lie group. Finally we will investigate a “linear independence” condition of representation rings, which are algebraic objects formed by representations of Lie groups. Students will have an opportunity to utilize such algebra packages as SAGE and Macaulay2 when carrying out the project.