Inequalities Involving Generalized Matrix Functions

Abstract

Given a set \(F\) of functions that have their domain and codomain in common, if the codomain is ordered we may partially order \(F\) by choosing a subset \(S\) of their domain and saying that for \(f,g \in F\), \(f \leq g\) means that \(f(x) \leq g(x)\) for every \(x \in S\). We call such an ordering a dominance ordering of \(F\) on \(S\). This dissertation is concerned with dominance orderings of (normalized) generalized matrix functions on the set of positive semidefinite matrices and on select subsets. We consider variants of Shur's theorem and Lieb's conjecture with different subsets of the group algebra and a different dominance order. In particular, we show that neither holds for Brauer characters with the usual ordering. We suggest a subset of the positive semidefinite matrices on which a modified Schur's theorem may hold for Brauer characters. It holds for Brauer characters of \(p\)-solvable groups and for many specific Brauer characters of symmetric groups. We provide a computational approach for generating dominance inequalities involving Brauer characters of symmetric groups.