# On Horn's Problem and its Volume Function

## TL;DR

This paper extends Horn's problem by analyzing the probability distribution of sums of elements from orbits of compact Lie groups, relating the volume function to geometric structures and tensor product multiplicities, with detailed examples.

## Contribution

It introduces a volume function for the sum of orbits, relates it to geometric and algebraic structures, and explores its properties and applications in representation theory.

## Key findings

- The volume function relates to symplectic and Riemannian geometry.
- Non-analyticities and vanishing of the volume function are discussed.
- In the coadjoint case, it computes volumes of convex polytopes linked to tensor product multiplicities.

## Abstract

We consider an extended version of Horn's problem: given two orbits $\mathcal{O}_\alpha$ and $\mathcal{O}_\beta$ of a linear representation of a compact Lie group, let $A\in \mathcal{O}_\alpha$, $B\in \mathcal{O}_\beta$ be independent and invariantly distributed random elements of the two orbits. The problem is to describe the probability distribution of the orbit of the sum $A+B$. We study in particular the familiar case of coadjoint orbits, and also the orbits of self-adjoint real, complex and quaternionic matrices under the conjugation actions of $\mathrm{SO}(n)$, $\mathrm{SU}(n)$ and $\mathrm{USp}(n)$ respectively. The probability density can be expressed in terms of a function that we call the volume function. In this paper, (i) we relate this function to the symplectic or Riemannian geometry of the orbits, depending on the case; (ii) we discuss its non-analyticities and possible vanishing; (iii) in the coadjoint case, we study its relation to tensor product multiplicities (generalized Littlewood--Richardson coefficients) and show that it computes the volume of a family of convex polytopes introduced by Berenstein and Zelevinsky. These considerations are illustrated by a detailed study of the volume function for the coadjoint orbits of $B_2=\mathfrak{so}(5)$.