The Horn problem and planar networks

TL;DR

This paper introduces a combinatorial approach using weighted planar graphs to characterize the eigenvalue set of sums of Hermitian matrices, connecting it to the Horn problem and related linear algebra problems.

Contribution

It presents a novel combinatorial framework that models the Horn problem and similar linear algebra problems via planar networks and triangular tableaux.

Findings

Equivalent polyhedral cone described by planar graph weights

Inner node values in tableaux have natural combinatorial interpretation

Framework applies to Gelfand--Zeitlin and other linear algebra problems

Abstract

The problem of determining the set of possible eigenvalues of 3 Hermitian matrices that sum up to zero is known as the Horn problem. The answer is a polyhedral cone, which, following Knutson and Tao, can be described as the projection of a simpler cone in the space of triangular tableaux (or hives) to the boundary nodes of the tableau. In this paper, we introduce a combinatorial problem defined in terms of certain weighted planar graphs giving rise to exactly the same polyhedral cone. In our framework, the values at the inner nodes of the triangular tableaux receive a natural interpretation. Other problems of linear algebra fit into the same scheme, among them the Gelfand--Zeitlin problem. Our approach is motivated by the works of Fomin and Zelevinsky on total positivity and by the ideas of tropicalization.