k-Schur Functions and Affine Schubert Calculus

  • Thomas Lam
  • Luc Lapointe
  • Jennifer Morse
  • Anne Schilling
  • Mark Shimozono
  • Mike Zabrocki
Part of the Fields Institute Monographs book series (FIM, volume 33)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, Mike Zabrocki
    Pages 1-7
  3. Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, Mike Zabrocki
    Pages 9-131
  4. Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, Mike Zabrocki
    Pages 133-168
  5. Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, Mike Zabrocki
    Pages 169-212
  6. Back Matter
    Pages 213-219

About this book

Introduction

This book gives an introduction to the very active field of combinatorics of affine Schubert calculus, explains the current state of the art, and states the current open problems. Affine Schubert calculus lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory. The other direction is the study of the Schubert bases of the (co)homology of the affine Grassmannian, an algebro-topological formulation of a problem in enumerative geometry.

This is the first introductory text on this subject. It contains many examples in Sage, a free open source general purpose mathematical software system, to entice the reader to investigate the open problems. This book is written for advanced undergraduate and graduate students, as well as researchers, who want to become familiar with this fascinating new field.

Keywords

Macdonald polynomial positivity Schubert bases Stanley symmetric functions affine Schubert calculus enumerative geometry representation theory

Authors and affiliations

  • Thomas Lam
    • 1
  • Luc Lapointe
    • 2
  • Jennifer Morse
    • 3
  • Anne Schilling
    • 4
  • Mark Shimozono
    • 5
  • Mike Zabrocki
    • 6
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Instituto de Matemática y FísicaUniversidad de TalcaTalcaChile
  3. 3.Department of MathematicsDrexel UniversityPhiladelphiaUSA
  4. 4.Department of MathematicsUniversity of CaliforniaDavisUSA
  5. 5.Department of MathematicsVirginia TechBlacksburgUSA
  6. 6.Mathematics and StatisticsYork UniversityTorontoCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4939-0682-6
  • Copyright Information Springer Science+Business Media New York 2014
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4939-0681-9
  • Online ISBN 978-1-4939-0682-6
  • Series Print ISSN 1069-5273
  • Series Online ISSN 2194-3079
  • About this book