Flag varieties : an interplay of geometry, combinatorics, and representation theory
著者
書誌事項
Flag varieties : an interplay of geometry, combinatorics, and representation theory
（Texts and readings in mathematics, 53）
Hindustan Book Agency, c2018
2nd ed
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注記
Includes bibliographical references (p. [295]-300) and index
内容説明・目次
内容説明
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.
目次
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
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