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arXiv · 1210.5003

Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

David Anderson·Edward Richmond·Alexander Yong
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Abstract

The saturation theorem of [Knutson-Tao '99] concerns the nonvanishing of Littlewood-Richardson coefficients. In combination with work of [Klyachko '98], it implies [Horn '62]'s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization [Friedland '00] to majorized sums of Hermitian matrices. 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 aformentioned work together with [Thomas-Yong '12].

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David Anderson, Edward Richmond, Alexander Yong. 2013-04-22. Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians. https://arxiv.org/abs/1210.5003

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