Majorization and Spherical Functions

TL;DR

This paper generalizes the concept of majorization using root systems and links it to spherical functions on Riemannian symmetric spaces, providing a new mathematical characterization.

Contribution

It introduces a generalized majorization framework associated with root systems and characterizes it via spherical functions on symmetric spaces.

Findings

Generalized majorization associated with root systems.

Characterization of this majorization through spherical functions.

Connections established between algebraic and geometric structures.

Abstract

Majorization is a partial order on real vectors which plays an important role in a variety of subjects, ranging from algebra and combinatorics to probability and statistics. In this paper, we consider a generalized notion of majorization associated to an arbitrary root system $\Phi,$ and show that it admits a natural characterization in terms of the values of spherical functions on any Riemannian symmetric space with restricted root system $\Phi.$