Honeycombs and sums of Hermitian matrices

TL;DR

This paper discusses the role of honeycombs and representation theory in understanding the spectra of sums of Hermitian matrices, highlighting recent advances in Horn's conjecture.

Contribution

It introduces honeycombs as a combinatorial tool and explores their connection to representation theory to analyze Hermitian matrix spectra.

Findings

Horn's conjecture has been affirmatively settled.

Honeycombs effectively encode spectral boundary conditions.

Connections to U(n) representation theory provide new insights.

Abstract

Horn's conjecture, which given the spectra of two Hermitian matrices describes the possible spectra of the sum, was recently settled in the affirmative. In this survey we discuss one of the many steps in this, which required us to introduce a combinatorial gadget called a {\em honeycomb}; the question is then reformulable as about the existence of honeycombs with certain boundary conditions. Another important tool is the connection to the representation theory of the group U(n), by ``classical vs. quantum'' analogies.