Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

E-ISSN： 1570-5846|149|9|1569-1582

ISSN： 0010-437x

Source： Compositio Mathematica, Vol.149, Iss.9, 2013-09, pp. : 1569-1582

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Abstract

The saturation theorem of Knutson and Tao concerns the nonvanishing of Littlewood–Richardson coefficients. In combination with work of Klyachko, it implies Horn’s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization to majorized sums of Hermitian matrices, due to S. Friedland. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland’s problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aforementioned work together with recent work of H. Thomas and A. Yong.