Flag Varieties: An Interplay of Geometry, Combinatorics, and Representation Theory (Texts and Readings in Mathematics Book 53)

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.-- Provided by publisher Preface Contents About the Authors Introduction 1 Preliminaries 1.1 Commutative Algebra 1.2 Affine Varieties 1.3 Projective Varieties 1.4 Schemes - Affine and Projective 1.5 The Scheme Spec(A) 1.6 The Scheme Proj(S) 1.7 Sheaves of OX-Modules 1.8 Attributes of Varieties 2 Structure Theory of Semisimple Rings 2.1 Semisimple Modules 2.2 Semisimple Rings 2.3 Brauer Groups and Central Simple Algebras 2.4 The Group Algebra, K[G] 2.5 The Center of K[G] Exercises 3 Representation Theory of Finite Groups 3.1 Representations of G 3.2 Characters of Representations 3.3 Ordinary Representations 3.4 Tensor Product of Representations 3.5 Contragradient Representations 3.6 Restrictions and Inductions 3.7 Character Group of G Exercises 4 Representation Theory of the Symmetric Group 4.1 The Symmetric Group Sn 4.2 Frobenius-Young Modules 4.3 Specht Modules 4.4 General Case 4.5 Proof of Theorem 4.3.7 4.6 Representation Theory of An Exercises 5 Symmetric Polynomials 5.1 Notation and Motivation 5.2 Several Bases for S (n) 5.3 Kostka Numbers & Determinantal Formulas 5.4 Results xλ Exercises 6 Schur-Weyl Duality and the Relationship Between Representations of Sd and GLn (C) 6.1 Generalities 6.2 Schur-Weyl Duality 6.3 Characters of the Schur Modules 6.4 Schur Module Representations of SLn (C) 6.5 Representations of GLn (C) Exercises 7 Structure Theory of Complex Semisimple Lie Algebras 7.1 Introduction to Semisimple Lie Algebras 7.2 The Exponential Map in Characteristic Zero 7.3 Structure of Semisimple Lie Algebras 7.4 Jordan Decomposition in Semisimple Lie Algebras 7.5 The Lie Algebra sln (C) 7.6 Cartan Subalgebras 7.7 Root Systems 7.8 Structure Theory of sln (C) Exercises 8 Representation Theory of Complex Semisimple Lie Algebras 8.1 Representations of g 8.2 Weight Spaces 8.3 Finite Dimensional Modules 8.4 Fundamental Weights 8.5 Dimension and Character Formulas 8.6 Irreducible sln (C)-Modules Exercises 9 Generalities on Algebraic Groups 9.1 Algebraic Groups and Their Lie Algebras 9.2 The Tangent Space 9.3 Jordan Decomposition in G 9.4 Variety Structure on G/H 9.5 The Flag Variety 9.6 Structure of Connected Solvable Groups 9.7 Borel Fixed Point Theorem 9.8 Variety of Borel Subgroups Exercises 10 Structure Theory of Reductive Groups 10.1 Cartan Subgroups 10.2 The Weyl Group 10.3 Regular and Singular Tori 10.4 Semisimple Rank 1 10.5 One Parameter Subgroups 10.6 Reductive Groups 10.7 Almost Simple Groups 10.8 Schubert Varieties & Bruhat Decomposition 10.9 Standard Parabolic Subgroups Exercises 11 Representation Theory of Semisimple Algebraic Groups 11.1 Weight Vectors 11.2 Geometric Realization of V(λ) 12 Geometry of the Grassmannian, Flag and their Schubert Varieties via Standard Monomial Theory 12.1 Grassmannian Variety 12.2 Gd,n Identified with a Homogeneous Space 12.3 Schubert Varieties 12.4 Standard Monomials 12.5 Equations Defining Schubert Varieties 12.6 Unions of Schubert Varieties 12.7 Vanishing Theorems 12.8 Results for F Exercises 13 Singular Locus of a Schubert Variety in the Flag Variety SLn/B 13.1 Generalities 13.2 Singular Loci of Schubert Varieties Exercises 14 Applications 14.1 Determinantal Varieties 14.2 Classical Invariant Theory 14.3 Toric Degeneration of a Schubert Variety Exercises 15 Free Resolutions of Some Schubert Singularities. 15.1 One geometric technique 15.2 Schubert varieties and homogeneous bundles 15.3 Properties of Schubert desingularization 15.4 Free resolutions 15.5 Cohomology of Homogeneous Vector-Bundles 15.6 Examples 15.7 Further remarks 16 Levi Subgroup Actions on Schubert Varieties, and Some Geometric Consequences 16.1 Preliminaries 16.2 Decomposition Results 16.3 Multiplicity Consequences of the Decomposition 16.4 Sphericity Consequences of the Decomposition 16.5 Singularities and the L-action in Degree 1 Appendix A Chevalley Groups A.1 A Basis for ε A.2 Kostant's Z-form A.3 Admissible Lattices A.4 The Chevalley Groups A.5 Simplicity of Groups of Adjoint Type A.6 Chevalley Groups and Algebraic Groups A.7 Generators, Relations of Universal Group References List of Symbols Index