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© 2015

The Grassmannian Variety

Geometric and Representation-Theoretic Aspects

Benefits

  • Presents an exciting playing field for graduate students and researchers in mathematics, physics, and computer science, to expand their knowledge in the field of algebraic geometry

  • Comprehensive treatment brings graduate students and new researchers to the forefront of the field

  • Serves as excellent supplementary material for a graduate course on Grassmannian varieties

Book

Part of the Developments in Mathematics book series (DEVM, volume 42)

Table of contents

  1. Front Matter
    Pages i-x
  2. V. Lakshmibai, Justin Brown
    Pages 1-3
  3. Algebraic Geometry: A Brief Recollection

    1. Front Matter
      Pages 5-5
    2. V. Lakshmibai, Justin Brown
      Pages 7-25
    3. V. Lakshmibai, Justin Brown
      Pages 27-38
    4. V. Lakshmibai, Justin Brown
      Pages 39-48
  4. Grassmann and Schubert Varieties

    1. Front Matter
      Pages 49-49
    2. V. Lakshmibai, Justin Brown
      Pages 51-71
    3. V. Lakshmibai, Justin Brown
      Pages 73-94
    4. V. Lakshmibai, Justin Brown
      Pages 95-113
  5. Flag Varieties and Related Varieties

    1. Front Matter
      Pages 115-115
    2. V. Lakshmibai, Justin Brown
      Pages 117-127
    3. V. Lakshmibai, Justin Brown
      Pages 129-142
    4. V. Lakshmibai, Justin Brown
      Pages 143-153
    5. V. Lakshmibai, Justin Brown
      Pages 155-162
  6. Back Matter
    Pages 163-172

About this book

Introduction

This book gives a comprehensive treatment of the Grassmannian varieties and their Schubert subvarieties, focusing on the geometric and representation-theoretic aspects of Grassmannian varieties. Research of Grassmannian varieties is centered at the crossroads of commutative algebra, algebraic geometry, representation theory, and combinatorics. Therefore, this  text uniquely presents an exciting playing field  for graduate students and researchers in mathematics, physics, and computer science, to expand their knowledge in the field of algebraic geometry. The standard monomial theory (SMT) for the Grassmannian varieties and their Schubert subvarieties are introduced and the text presents some important applications of SMT including the Cohen–Macaulay property, normality, unique factoriality, Gorenstein property, singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory.

This text would serve well as a reference book for a graduate work on Grassmannian varieties and would be an excellent supplementary text for several courses including those in geometry of spherical varieties, Schubert varieties, advanced topics in geometric and differential topology, representation theory of compact and reductive groups, Lie theory, toric varieties, geometric representation theory, and singularity theory.​ The reader should have some familiarity with commutative algebra and algebraic geometry.

Keywords

Bott-Samelson varieties Cohen-Macaulay property Flag varieties Geometry of spherical varieties Gorenstein property Groebner bases Schubert varieties Singular locus Standard monomial theory The Grassmannian variety algebraic geometry classical invariant theory

Authors and affiliations

  1. 1.Department of MathematicsNortheastern UniversityBostonUSA
  2. 2.Department of MathematicsOlivet Nazarene UniversityBourbonnaisUSA

Bibliographic information

  • Book Title The Grassmannian Variety
  • Book Subtitle Geometric and Representation-Theoretic Aspects
  • Authors V. Lakshmibai
    Justin Brown
  • Series Title Developments in Mathematics
  • Series Abbreviated Title DEVM
  • DOI https://doi.org/10.1007/978-1-4939-3082-1
  • Copyright Information Springer Science+Business Media New York 2015
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-1-4939-3081-4
  • Softcover ISBN 978-1-4939-5608-1
  • eBook ISBN 978-1-4939-3082-1
  • Series ISSN 1389-2177
  • Series E-ISSN 2197-795X
  • Edition Number 1
  • Number of Pages X, 172
  • Number of Illustrations 84 b/w illustrations, 39 illustrations in colour
  • Topics Algebraic Geometry
    Algebraic Topology
    Combinatorics
  • Buy this book on publisher's site
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Reviews

“The present book gives a detailed treatment of the standard monomial theory (SMT) for the Grassmannians and their Schubert subvarieties along with several applications of SMT. It can be used as a reference book by experts and graduate students who study varieties with a reductive group action such as flag and toric varieties.” (Valentina Kiritchenko, zbMATH 1343.14001, 2016)

“The book under review is more elementary; it is exclusively devoted to Grassmannians and their Schubert subvarieties. The book is divided into three parts. … This is a nicely written book, one that may appeal to students and researchers in related areas.” (Felipe Zaldivar, MAA Reviews, maa.org, December, 2015)