Path Tableaux and the Combinatorics of the Immanant Function

Abstract

Immanants are a generalization of the well-studied determinant and permanent. Although the combinatorial interpretations for the determinant and permanent have been studied in excess, there remain few combinatorial interpretations for the immanant.

The main objective of this thesis is to consider the immanant, and its possible combinatorial interpretations, in terms of recursive structures on the character. This thesis presents a comprehensive view of previous interpretations of immanants. Furthermore, it discusses algebraic techniques that may be used to investigate further into the combinatorial aspects of the immanant.

We consider the Temperley-Lieb algebra and the class of immanants over the elements of this algebra. Combinatorial tools including the Temperley-Lieb algebra and Kauffman diagrams will be used in a number of interpretations. In particular, we extend some results for the permanent and determinant based on the $R$-weighted planar network construction, where $R$ is a convenient ring, by Clearman, Shelton, and Skandera. This thesis also presents some cases in which this construction cannot be extended. Finally, we present some extensions to combinatorial interpretations on certain classes of tableaux, as well as certain classes of matrices.