William Fulton

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This is a closed subscheme ofX ; locally, where the bundles are trivial, this is defined by vanishing of the minors of size rij +1 in the product of matrices giving the map φj ◦ · · · ◦ φi+1 from Ei to Ej , for all i < j. Not all rank conditions give reasonable loci. Those that do—and the only ones we will consider—are characterized by the conditions rij ≤(More)
The aim of this paper is to introduce some polynomials that specialize to all previously known Schubert polynomials: the classical Schubert polynomials of Lascoux and Schützenberger [L-S], [M], the quantum Schubert polynomials of Fomin, Gelfand, and Postnikov [F-G-P], and quantum Schubert polynomials for partial flag varieties of Ciocan-Fontanine [CF2].(More)
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 Peterson’s quantum version of Chevalley’s formula, also for general G/P ’s.
Under appropriate conditions on the rank function r, which guarantee that, for generic h, f,(h) is irreducible, we prove a formula for the class [f,(h)] of this locus in the Chow or cohomology ring of X, as a polynomial in the Chern classes of the vector bundles. When expressed in terms of Chern roots, these polynomials are the "double Schubert polynomials"(More)
Answering a question raised by S. Friedland, we show that the possible eigenvalues of Hermitian matrices (or compact operators) A, B, and C with C ≤ A+B are given by the same inequalities as in Klyachko’s theorem for the case where C = A + B, except that the equality corresponding to tr(C) = tr(A) + tr(B) is replaced by the inequality corresponding to tr(C)(More)
This paper reports results from a detailed travel diary survey of 2125 residents in the South Bay area of Los Angeles County - a mature, auto-oriented suburban region. Study areas were divided into four centres, typical of compact development or smart growth, and four linear, auto-oriented corridors. Results show substantial variation in the amount of(More)
The purpose of this exposition is to give a simple treatment of Knutson and Tao’s recent proof of the saturation conjecture [10]. A finite dimensional irreducible polynomial representation of GLn(C) is determined by its highest weight, which is a weakly decreasing sequence of n non-negative integers, also called a partition [5, §8]. The irreducible(More)