  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.
