Additive Eigenvalue Problem (a survey), (With appendix by M. Kapovich)

TL;DR

This survey reviews the classical and extended Hermitian eigenvalue problems, highlighting the development of linear inequalities characterizing eigenvalues of matrix sums and their generalizations to complex algebraic groups.

Contribution

It presents a comprehensive overview of the additive eigenvalue problem, including the authors' new optimal inequalities using a deformed cohomology product for any semisimple group.

Findings

Classical eigenvalue inequalities are fully characterized by linear inequalities.

The authors introduce a deformation of the cohomology product to obtain optimal inequalities.

The survey includes detailed proofs and extends the problem to complex algebraic groups.

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 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 an optimal set of inequalities for the problem in this case. The Hermitian eigenvalue problem has been extended by Berenstein-Sjamaar (2000) and Kapovich-Leeb-Millson (2005) 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 flag 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.