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Fonctions Symétriques, Polynômes de Schubert et Lieux de Dégénérescence
Laurent Manivel, Université de Grenoble I, Institut Fourier, Saint Martin d'Hères Cedex, France
A publication of the Société Mathématique de France.
Cours Spécialisés--Collection SMF
1998; 179 pp; softcover
Number: 3
ISBN-10: 2-85629-066-3
ISBN-13: 978-2-85629-066-8
List Price: US$44
Member Price: US$35.20
Order Code: COSP/3
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This course begins with two chapters of combinatorial nature. The first is devoted to symmetric functions and to the properties of Schur polynomials, studied using Young tableaux and the Knuth insertion algorithm. It is shown that these polynomials can be identified with the characters of the irreducible representations of the symmetric group.

The second chapter is a study of Schubert polynomials, as defined by A. Lascoux and M.-P. Schützenberger in terms of divided differences. These polynomials are associated with permutations. Their combinatorics is related to the Bruhat order on symmetric groups and to certain Hecke algebras of these groups.

The third and final chapter is of geometrical nature. Its main theme is the study of Schubert varieties inside Grassmannians and flag manifolds. The fact that the homology classes of these varieties can be represented by Schur or Schubert polynomials allows geometrical translation of most of the results of the first two chapters. And since these Schubert varieties are universal models for certain degeneracy loci of morphisms between vector bundles, expressions are deduced for the homology classes of these loci in terms of characteristic classes of the bundles involved. Text is in French.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians working in combinatorics.

Table of Contents

  • Introduction
  • L'anneau des fonctions symétriques
  • Les polynômes de Schubert
  • Les variétés de Schubert
  • Une brève introduction à l'homologie singulière
  • Bibliographie
  • Index
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