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

Ressayre, Nicolas

Int. Math. Res. Not. 2012, No. 21, 4966-5005 (2012).

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.

Reviewer: Richárd Rimányi (Chapel Hill)

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

Keywords:

Horn conjecture; eigencone; cohomology-free description

Citations:

Zbl 1258.14008

Software:

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