Flag varieties : an interplay of geometry, combinatorics, and representation theory

Author(s)

• Lakshmibai, V. (Venkatramani)
• Brown, Justin (Justin Allen)

Bibliographic Information

Flag varieties : an interplay of geometry, combinatorics, and representation theory

V. Lakshmibai, Justin Brown

（Texts and readings in mathematics, 53）

Hindustan Book Agency, c2018

2nd ed

Note

Includes bibliographical references (p. [295]-300) and index

Description and Table of Contents

Description

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 presents a discussion of complex semisimple Lie algebras and of semisimple algebraic groups; in addition, the representation theory of symmetric groups is also discussed. In the area of algebraic geometry, the book gives a detailed account of Grassmann varieties, flag varieties, and their Schubert subvarieties. Because of their 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. In this second edition, two recent results on Schubert varieties in the Grassmannian have been added, and some errors in the first edition corrected.

Table of Contents

Preface Introduction 1 Preliminaries 2 Structure Theory of Semisimple Rings 3 Representation Theory of Finite Groups 4 Representation Theory of the Symmetric Group 5 Symmetric Polynomials 6 Schur-Weyl Duality and the Relationship Between Representations of Sd and GLn (C) 7 Structure Theory of Complex Semisimple Lie Algebras 8 Representation Theory of Complex Semisimple Lie Algebras 9 Generalities on Algebraic Groups 10 Structure Theory of Reductive Groups 11 Representation Theory of Semisimple Algebraic Groups 12 Geometry of the Grassmannian, Flag and their Schubert Varieties via Standard Monomial Theory 13 Singular Locus of a Schubert Variety in the Flag Variety SLn / B 14 Applications 15 Free Resolutions of Some Schubert Singularities 16 Levi Subgroup Actions on Schubert Varieties, and Some Geometric Consequences Appendix A: Chevalley Groups References List of Symbols Index

by "Nielsen BookData"

Details

• NCID: BB26289644
• ISBN:
  • 9789386279705
• Country Code: ii
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: New Delhi
• Pages/Volumes: xiii, 310 p.
• Size: 25 cm
• Classification:
  • DC20 : 516.35
• Subject Headings:
  • LCSH : Geometry, Algebraic
  • LCSH : Flag manifolds
  • LCSH : Representations of groups
  • LCSH : Semisimple Lie groups
  • LCSH : Schubert varieties
• Parent Bibliography ID:
  • BA21367153