The symplectic and algebraic geometry of Horn's problem

Horn's problem was the following: given two Hermitian matrices with known spectra, what might be the eigenvalue spectrum of the sum? This linear algebra problem is exactly of the sort to be approached with the methods of modern Hamiltonian geometry (which were unavailable to Horn). The theorem linking symplectic quotients and geometric invariant theory lets one also bring algebraic geometry and representation theory into play. This expository note is intended to elucidate these connections for linear algebraists, in the hope of making it possible to recognize what sort of problems are likely to fall to the same techniques that were used in proving Horn's conjecture.Comment: 16 pages, 1 figure; expository conference paper (second version has inessential cosmetic changes

Schubert calculus and shifting of interval positroid varieties

Consider k x n matrices with rank conditions placed on intervals of columns. The ranks that are actually achievable correspond naturally to upper triangular partial permutation matrices, and we call the corresponding subvarieties of Gr(k,n) the _interval positroid varieties_, as this class lies within the class of positroid varieties studied in [Knutson-Lam-Speyer]. It includes Schubert and opposite Schubert varieties, and their intersections, and is Grassmann dual to the projection varieties of [Billey-Coskun]. Vakil's "geometric Littlewood-Richardson rule" [Vakil] uses certain degenerations to positively compute the H^*-classes of Richardson varieties, each summand recorded as a (2+1)-dimensional "checker game". We use his same degenerations to positively compute the K_T-classes of interval positroid varieties, each summand recorded more succinctly as a 2-dimensional "K-IP pipe dream". In Vakil's restricted situation these IP pipe dreams biject very simply to the puzzles of [Knutson-Tao]. We relate Vakil's degenerations to Erd\H os-Ko-Rado shifting, and include results about computing "geometric shifts" of general T-invariant subvarieties of Grassmannians.Comment: 35 pp; this subsumes and obviates the unpublished http://arxiv.org/abs/1008.430