Polyhedral Analysis of Plethysms and Kronecker Coefficients

Abstract

We study three types of polytopes occurring in combinatorial representation theory. The first two are related to Kronecker coefficients; i.e. the tensor product multiplicities for the irreducible representations of the symmetric group. The third is related to the plethystic representations of GL(n) formed by composing Schur functors. Recently, Stembridge generalized a stabilization phenomenon occurring among the Kronecker coefficients that was first discovered by Murnaghan in the 1930's. While doing so, Stembridge demonstrated that this generalized stability will occur when either of two associated polytopes is 0-dimensional. In this thesis, we will show how to classify when either of these polytopes is 0-dimensional by studying a related transportation polytope for the first polytope and a modified version of the puzzles and honeycombs of Knutson and Tao for the second polytope. Finally, we study the weight polytopes of plethystic representations. These polytopes are the convex hull of all weights of the associated representation. We construct an algorithm for finding the vertices and a set of defining inequalities for this polytope, and use it to identify weights of the representation that are maximal with respect to the dominance order. Surprisingly, not all integer points in the weight polytope are weights of the representation.