The Multiset Partition Algebra: Diagram-Like Bases and Representations

There is a classical connection between the representation theory of the symmetric group and the general linear group called Schur--Weyl Duality. Variations on this principle yield analogous connections between the symmetric group and other objects such as the partition algebra and more recently the multiset partition algebra. The partition algebra has a well-known basis indexed by graph-theoretic diagrams which allows the multiplication in the algebra to be understood visually as combinations of these diagrams. My thesis begins with a construction of an analogous basis for the multiset partition algebra. It continues with applications of this basis to constructing the irreducible representations of the multiset partition algebra and analysis of subalgebras analogous to the Tanabe algebras and the Brauer algebra. Finally, I address connections between the multiset partition algebra and longstanding questions in the representation theory of the symmetric group including the Kronecker problem and the restriction problem from GL_n to S_n.