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Eigenvalues, invariant factors, highest weights, and Schubert calculus. (English) Zbl 0994.15021
Summary: We describe recent work of A. A. Klyachko [Sel. Math., New Ser. 4, No. 3, 419-445 (1998; Zbl 0915.14010)], B. Totaro [Geometry and analysis on complex manifolds. 242-250 (1994; Zbl 0873.14016)], A. Knutson and T. Tao [J. Am. Math. Soc. 12, NO. 4, 1055-1090 (1999; Zbl 0944.05057)] that characterizes eigenvalues of sums of Hermitian matrices and decomposition of tensor products of representations of \(GL_{n}(\mathbb{C})\). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices.

MSC:
15A42 Inequalities involving eigenvalues and eigenvectors
14M15 Grassmannians, Schubert varieties, flag manifolds
05E15 Combinatorial aspects of groups and algebras (MSC2010)
13F10 Principal ideal rings
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
15A18 Eigenvalues, singular values, and eigenvectors
47B07 Linear operators defined by compactness properties
22E46 Semisimple Lie groups and their representations
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