Subgroup Majorization

TL;DR

This paper extends the concept of majorization to more general groups beyond permutations, focusing on subgroups and their implications for order-preserving functions, with a key example involving the hyperoctahedral group.

Contribution

It advances the theory of G-majorization by exploring subgroup and quotient group extensions, and analyzes their effects on fundamental cones and order-preserving functions.

Findings

Extended majorization theory to subgroups and quotient groups.

Analyzed implications for fundamental cones and order-preserving functions.

Provided detailed study of hyperoctahedral group actions.

Abstract

The extension of majorization (also called the rearrangement ordering), to more general groups than the symmetric (permutation) group, is referred to as $G$-majorization. There are strong results in the case that $G$ is a reflection group and this paper builds on this theory in the direction of subgroups, normal subgroups, quotient groups and extensions. The implications for fundamental cones and order-preserving functions are studied. The main example considered is the hyperoctahedral group, which, acting on a vector in $\mathbb R^n$, permutes and changes the signs of components.