Eigenvalues, invariant factors, highest weights, and Schubert calculus. (English) Zbl 0994.15021

Fulton, William

Bull. Am. Math. Soc., New Ser. 37, No. 3, 209-249 (2000).

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

Keywords:

eigenvalues; sums of Hermitian matrices; decomposition of tensor products; representationof \(GL_n(\mathbb{C})\); invariant factors; products of matrices; Grassmannian varieties; singular

Citations:

Zbl 0915.14010; Zbl 0873.14016; Zbl 0944.05057

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