Eigenvalues, invariant factors, highest weights, and Schubert calculus

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