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A cohomology-free description of eigencones in types A, B, and C. (English) Zbl 1270.14026

The Horn conjecture is concerned with the possible eigenvalues of triples of Hermitian matrices that add up to the 0-matrix. A natural generalization is considering the triples of adjoint orbits of a compact connected Lie group, such that the sum of the orbits contain 0. The set of these triples parameterize a closed convex polyhedral cone. The challenge is to find a (minimal) set of inequalities defining this cone. This has been achieved by works of Klyachko, Knutson, Tao, Woodward, Belkale, Kumar and others. The first achievement of the paper is a description of a minimal set of inequalities defined inductively and without using cohomology. The second part of the paper deals with generalizations to connected reductive complex groups. For this case earlier works of P. Belkale, Sh. Kumar and N. Ressayre (see, e.g., [Math. Ann. 354, No. 2, 401–425 (2012; Zbl 1258.14008)]) provide a minimal set of inequalities. In the present paper the key condition of the Belkale-Kumar description is showed to be equivalent with the fact that two ordinary LR coefficients are both 1. The author derives this theorem from his cohomology free description.

MSC:

14N15 Classical problems, Schubert calculus
14M15 Grassmannians, Schubert varieties, flag manifolds
15A18 Eigenvalues, singular values, and eigenvectors
15B57 Hermitian, skew-Hermitian, and related matrices
17B99 Lie algebras and Lie superalgebras

Citations:

Zbl 1258.14008
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