The Horn inequalities from a geometric point of view

TL;DR

This paper provides a clear, geometric exposition of the Horn inequalities, linking tensor invariants, eigenvalues, and Schubert intersections, with a focus on accessible proofs and computational relevance.

Contribution

It offers a self-contained, geometric approach to Horn inequalities without relying on complex combinatorial rules, enhancing understanding and computational methods.

Findings

Geometric interpretation of Horn inequalities

Connection between tensor invariants and Schubert intersections

Accessible proofs without Littlewood-Richardson rule

Abstract

We give an exposition of the Horn inequalities and their triple role characterizing tensor product invariants, eigenvalues of sums of Hermitian matrices, and intersections of Schubert varieties. We follow Belkale's geometric method, but assume only basic representation theory and algebraic geometry, aiming for self-contained, concrete proofs. In particular, we do not assume the Littlewood-Richardson rule nor an a priori relation between intersections of Schubert cells and tensor product invariants. Our motivation is largely pedagogical, but the desire for concrete approaches is also motivated by current research in computational complexity theory and effective algorithms.