Eigenvalues, Invariant Factors, Highest Weights, and Schubert Calculus

We describe recent work of Klyachko, Totaro, Knutson, and Tao that characterizes eigenvalues of sums of Hermitian matrices and decomposition of tensor products of representations of GLn(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. Recent breakthroughs, primarily by A. Klyachko, B. Totaro, A. Knutson, and T. Tao, with contributions by P. Belkale and C. Woodward, have led to complete solutions of several old problems involving the various notions in the title. Our aim here is to describe this work and especially to show how these solutions are derived from it. Along the way, we will see that these problems are also related to other areas of mathematics, including geometric invariant theory, symplectic geometry, and combinatorics. In addition, we present some related applications to singular values of arbitrary matrices. Although many of the theorems we state here have not appeared elsewhere, their proofs are mostly “soft” algebra based on the hard geometric or combinatorial work of others. Indeed, this paper emphasizes concrete elementary arguments. We do give some new examples and counterexamples and raise some new open questions. We have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before. However, there is a very large literature, particularly for linear algebra problems about eigenvalues, singular values, and invariant factors. We have listed only a few of these articles, from whose bibliographies, we hope, an interested reader can trace the history; we apologize to the many whose work is not cited directly. We begin in the first five sections by describing each of the problems, together with some of their early histories, and we state as theorems the new solutions to these problems. In Section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs. Then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians. Sections 7, 8, 9, and 10 also contain variations and generalizations of some of the theorems stated in the first five sections, as well as attributions of the theorems to their authors. One of our fascinations with this subject, even now that we have proofs of the theorems, is the challenge to understand in a deeper way why all these subjects are Received by the editors in July 1999 and in revised form January 3, 2000. 2000 Mathematics Subject Classification. Primary 15A42, 22E46, 14M15; Secondary 05E15, 13F10, 14C17, 15A18, 47B07. The author was partly supported by NSF Grant #DMS9970435. c ©2000 American Mathematical Society