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Flag Varieties: An Interplay of Geometry, Combinatorics, and Representation Theory
V. Lakshmibai and Justin Brown, Northeastern University, Boston, MA
A publication of Hindustan Book Agency.
Hindustan Book Agency
2009; 288 pp; hardcover
ISBN-10: 81-85931-92-5
ISBN-13: 978-81-85931-92-0
List Price: US$52
Member Price: US$41.60
Order Code: HIN/40
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Flag varieties are important geometric objects and their study involves an interplay of geometry, combinatorics, and representation theory. This book is a detailed account of this interplay. In the area of representation theory, the book discusses complex semisimple Lie algebras and semisimple algebraic groups; in addition, the representation theory of symmetric groups is discussed. In the area of algebraic geometry, the book explains in detail Grassmannian varieties, flag varieties, and their Schubert subvarieties.

Because of the connections with root systems, many of the geometric results admit elegant combinatorial description, a typical example being the description of the singular locus of a Schubert variety. This is shown to be a consequence of standard monomial theory (abbreviated SMT). Thus the book includes SMT and some important applications--singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory.

A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels.


Graduate students and research mathematicians interested in flag varieties.

Table of Contents

  • Introduction
  • Preliminaries
  • Structure theory of semisimple rings
  • Representation theory of finite groups
  • Representation theory of the symmetric group
  • Symmetric polynomials
  • Schur-Weyl duality and the relationship between representations of \(Sd\) and \(GL_n (C)\)
  • Structure theory of complex semisimple Lie algebras
  • Representation theory of complex semisimple Lie algebras
  • Generalities on algebraic groups
  • Structure theory of reductive groups
  • Representation theory of semisimple algebraic groups
  • Geometry of the grassmannian, flag and their Schubert varieties via standard monomial theory
  • Singular locus of a Schubert variety in the flag variety \(SL_n/B\)
  • Applications
  • Appendix: Chevalley groups
  • Bibliography
  • List of symbols
  • Index
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