General Linear Group and Symmetric Group: Commuting Actions and Combinatorics (e-copy ONLY) ...[e-MSc.-1]

Abstract

In this thesis, we have explored the representation theories of two prototypical examples of finite and infinite groups, the symmetric group and the general linear group, over the base field of complex numbers. More specifically, we are interested in understanding the connection between these two groups’ representations and seeing the ramifications they have on each other, while trying to make the exposition combinatorial in nature all the while. Robinson-Schensted-Knuth correspondence and its dual have been employed to deduce many character identities throughout, which in turn yield nontrivial facts about representations. After discussing concrete realizations of irreducible representations of these two groups and establishing the bridge between these worlds, we use this machinery to go back and forth, which in turn shed new lights on Gelfand models of symmetric groups. Finally, we use SAGE computations to work out concrete answer to a naturally motivated question we raised in this thesis.