A SURVEY OF THE ADDITIVE EIGENVALUE PROBLEM (WITH APPENDIX BY M. KAPOVICH)

Abstract

The classical Hermitian eigenvalue problem addresses the following question: What are the possible eigenvalues of the sum A + B of two Hermitian matrices A and B, provided we fix the eigenvalues of A and B. A systematic study of this problem was initiated by H. Weyl (1912). By virtue of contributions from a long list of mathematicians, notably Weyl (1912), Horn (1962), Klyachko (1998) and Knutson–Tao (1999), the problem is finally settled. The solution asserts that the eigenvalues of A + B are given in terms of certain system of linear inequalities in the eigenvalues of A and B. These inequalities (called the Hom inequalities) are given explicitly in terms of certain triples of Schubert classes in the singular cohomology of Grassmannians and the standard cup product. Belkale (2001) gave a smaller set of inequalities for the problem in this case (which was shown to be optimal by Knutson–Tao–Woodward). The Hermitian eigenvalue problem has been extended by Berenstein–Sjamaar (2000) and Kapovich–Leeb–Millson (2009) for any semisimple complex algebraic group G. Their solution is again in terms of a system of linear inequalities obtained from certain triples of Schubert classes in the singular cohomology of the partial ag varieties G/P (P being a maximal parabolic subgroup) and the standard cup product. However, their solution is far from being optimal. In a joint work with P. Belkale, we define a deformation of the cup product in the cohomology of G/P and use this new product to generate our system of inequalities which solves the problem for any G optimally (as shown by Ressayre). This article is a survey (with more or less complete proofs) of this additive eigenvalue problem. The eigenvalue problem is equivalent to the saturated tensor product problem. We also give an extension of the saturated tensor product problem to the saturated restriction problem for any pair G ⊂ Ĝ of connected reductive algebraic groups. In the appendix by M. Kapovich, a connection between metric geometry and the representation theory of complex semisimple algebraic groups is explained. The connection runs through the theory of buildings. This connection is exploited to give a uniform (though not optimal) saturation factor for any G.