# Revisiting Horn's Problem

## TL;DR

This paper reviews recent advances in Horn's problem, focusing on eigenvalue spectra of matrix sums, generalizations involving Lie group actions, and connections to representation theory and combinatorics.

## Contribution

It synthesizes recent results on probability measures, orbital integrals, and singularities in Horn's problem, and introduces new insights related to Schur's problem.

## Key findings

- Progress in computing probability densities via orbital integrals
- Identification of singularity loci in eigenvalue distributions
- Connections established between Horn's problem and representation theory

## Abstract

We review recent progress on Horn's problem, which asks for a description of the possible eigenspectra of the sum of two matrices with known eigenvalues. After revisiting the classical case, we consider several generalizations in which the space of matrices under study carries an action of a compact Lie group, and the goal is to describe an associated probability measure on the space of orbits. We review some recent results about the problem of computing the probability density via orbital integrals and about the locus of singularities of the density. We discuss some relations with representation theory, combinatorics, pictographs and symmetric polynomials, and we also include some novel remarks in connection with Schur's problem.