Combinatorics of Transition Matrices of Symmetric and Polysymmetric Functions

Abstract

Symmetric functions are formal power series of bounded degree that are invariant under every permutation of their variables. The set $sym$ of symmetric functions has a $QQ$-algebra structure, and the transition matrices between bases of $sym$ encode combinatorial and representation-theoretic information. The study of transition matrices of {symmetric functions} is an active research program with a long illustrious history stretching back to work of Newton and Girard. In this thesis, we contribute to this research program in two broad ways. First, we focus on a new generalization of $sym$ introduced by Asvin G and Andrew O'Desky called the algebra of {polysymmetric functions} ($psym$). $psym$ can be viewed as a tensor product of copies of Sym where the variables in the $i$th copy are scaled by $i$. One method to construct a linear basis of $psym$ is to consider scaled pure-tensor products of a chosen basis of $sym$. In their paper, G and O'Desky define families of bases not arising from pure-tensors which we call {plethystic bases}. We prove combinatorial interpretations for transition matrices involving plethystic and pure-tensor bases via bijective methods. Second, we describe a general {combinatorial framework} which proves via local involutions that two transition matrices are mutually inverse. We apply this framework to bijectively prove the mutual invertibility of Kostka and inverse Kostka matrices, the orthogonality of symmetric group characters, the M"{o}bius inversion formula on the refinement lattice of compositions, and various inversion results involving brick tabloids.