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Flag Varieties

An Interplay of Geometry, Combinatorics, and Representation Theory

  • V. Lakshmibai
  • Justin Brown
Book

Part of the Texts and Readings in Mathematics book series (TRIM, volume 53)

Table of contents

  1. Front Matter
    Pages i-xv
  2. V. Lakshmibai, Justin Brown
    Pages 1-19
  3. V. Lakshmibai, Justin Brown
    Pages 21-38
  4. V. Lakshmibai, Justin Brown
    Pages 39-48
  5. V. Lakshmibai, Justin Brown
    Pages 49-68
  6. V. Lakshmibai, Justin Brown
    Pages 69-77
  7. V. Lakshmibai, Justin Brown
    Pages 89-101
  8. V. Lakshmibai, Justin Brown
    Pages 103-114
  9. V. Lakshmibai, Justin Brown
    Pages 115-133
  10. V. Lakshmibai, Justin Brown
    Pages 135-151
  11. V. Lakshmibai, Justin Brown
    Pages 153-163
  12. V. Lakshmibai, Justin Brown
    Pages 187-192
  13. V. Lakshmibai, Justin Brown
    Pages 193-201
  14. V. Lakshmibai, Justin Brown
    Pages 203-229
  15. Back Matter
    Pages 285-312

About this book

Introduction

This book discusses the importance of flag varieties in geometric objects and elucidates its richness as interplay of geometry, combinatorics and representation theory. The book presents a discussion on the representation theory of complex semisimple Lie algebras, as well as the representation theory of semisimple algebraic groups. In addition, the book also discusses the representation theory of symmetric groups. In the area of algebraic geometry, the book gives a detailed account of the Grassmannian varieties, flag varieties, and their Schubert subvarieties. Many of the geometric results admit elegant combinatorial description because of the root system connections, a typical example being the description of the singular locus of a Schubert variety. This discussion is carried out as a consequence of standard monomial theory. Consequently, this book includes standard monomial theory and some important applications—singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory. The two recent results on Schubert varieties in the Grassmannian have also been included in this book. The first result gives a free resolution of certain Schubert singularities. The second result is about certain Levi subgroup actions on Schubert varieties in the Grassmannian and derives some interesting geometric and representation-theoretic consequences.

Keywords

Semisimple Rings Finite Groups Symmetric Group Symmetric Polynomials Schur-Weyl Duality Semisimple Lie Algebras Algebraic Groups Reductive Groups

Authors and affiliations

  • V. Lakshmibai
    • 1
  • Justin Brown
    • 2
  1. 1.Northeastern UniversityBostonUSA
  2. 2.Olivet Nazarene UniversityBourbonnaisUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-981-13-1393-6
  • Copyright Information Springer Nature Singapore Pte Ltd. 2018 and Hindustan Book Agency 2018 2018
  • Publisher Name Springer, Singapore
  • eBook Packages Mathematics and Statistics
  • Online ISBN 978-981-13-1393-6
  • Series Print ISSN 2366-8717
  • Series Online ISSN 2366-8725
  • Buy this book on publisher's site