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abstract: 'We give a simplification of Belkale’s geometric proof of the Horn conjecture. Our approach uses the geometry of two-step flag manifolds to explain the occurrence of the Horn inequalities in a very straightforward way. The arguments for both necessity and sufficiency of the Horn inequalities are fairly conceptual when viewed in this framework. We provide examples to illustrate the method of proof.'
author:
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    Kevin Purbhoo\
    University of British Columbia
title: 'Two-step flag manifolds and the Horn conjecture'
---

Introduction
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General approach
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Horn’s conjecture [@H] was originally formulated as a recursive method for solving a problem concerning the eigenvalues of Hermitian matrices. However, as a consequence of work of Klyachko [@Kly], Horn’s conjecture can be reformulated as saying that the non-vanishing Schubert intersection numbers for Grassmannians satisfy a certain recursion. This recursion was first proved by Knutson and Tao [@KT], using combinatorial methods. Later, a geometric proof was given by Belkale [@B], which was the inspiration for this paper.

For our purposes, a Schubert problem will refer to a collection of Schubert varieties $\bar{\Omega}_{\sigma_1}, \ldots, 
\bar{\Omega}_{\sigma_s}$, on some partial flag variety $G/P$. A Schubert problem is non-vanishing if the product of the corresponding cohomology classes 