William Fulton

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

Riemann-Roch for Singular Varieties

The basic tool for a general Riemann-Roch theorem is MacPherson’s graph construction, applied to a complex E. of vector bundles on a scheme Y, exact off a closed subset X. This produces a localized Chern character1 ch x y (E.) which lives in the bivariant group \( A{\left( {X \to Y} \right)_\mathbb{Q}} \) For each class α∈A * Y, this gives a class

Intersection theory on toric varieties

ch_X^Y\left( {E.} \right) \cap \alpha \in {A_ * }{X_\mathbb{Q}}

Rational equivalence on singular varieties

whose image in \( {A_ * }{Y_\mathbb{Q}} \) is \( {\sum {\left( { - 1} \right)} ^i}ch\left( {{E_i}} \right) \cap \alpha \) The properties needed for Riemann-Roch, in particular the invariance under rational deformation, follow from the bivariant nature of ch x y E.

On the Fundamental Group of the Complement of a Node Curve

Introduction Let C be a smooth complex projective curve of genus g, and let J be the Jacobian of C. Upon choosing a base-point in C, J may be identified with the set of linear equivalence classes of divisors of degree d on C. Denote by W~ the algebraic subvariety of J parametrizing divisors which move in a linear system of dimension at least r. A fundamental theorem of Kempf [9] and Kleiman and Laksov [11, 12] asserts tha t these loci are nonempty when their expected dimension

Chern class formulas for quiver varieties

Abstract The operational Chow cohomology classes of a complete toric variety are identified with certain functions, called Minkowski weights, on the corresponding fan. The natural product of Chow cohomology classes makes the Minkowski weights into a commutative ring; the product is computed by a displacement in the lattice, which corresponds to a deformation in the toric variety. We show that, with rational coefficients, this ring embeds in McMullens polytope algebra, and that the polytope algebra is the direct limit of these Chow rings, over all compactifications of a given torus. In the nonsingular case, the Minkowski weight corresponding to the Todd class is related to a certain Ehrhart polynomial.

Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients

We give a formula for the smallest powers of the quantum parameters q that occur in a product of Schubert classes in the (small) quantum cohomology of general flag varieties G/P. We also include a complete proof of Petersons quantum version of Chevalleys formula, also for general G/Ps.

Universal Schubert polynomials

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