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 6 Contents 8 About the Authors 12 Introduction 13 1 Preliminaries 16 1.1 Commutative Algebra 16 1.2 Affine Varieties 20 1.3 Projective Varieties 24 1.4 Schemes - Affine and Projective 25 1.5 The Scheme Spec(A) 27 1.6 The Scheme Proj(S) 28 1.7 Sheaves of OX-Modules 29 1.8 Attributes of Varieties 33 2 Structure Theory of Semisimple Rings 35 2.1 Semisimple Modules 35 2.2 Semisimple Rings 40 2.3 Brauer Groups and Central Simple Algebras 46 2.4 The Group Algebra, K[G] 48 2.5 The Center of K[G] 49 Exercises 51 3 Representation Theory of Finite Groups 53 3.1 Representations of G 53 3.2 Characters of Representations 54 3.3 Ordinary Representations 57 3.4 Tensor Product of Representations 58 3.5 Contragradient Representations 59 3.6 Restrictions and Inductions 59 3.7 Character Group of G 61 Exercises 61 4 Representation Theory of the Symmetric Group 63 4.1 The Symmetric Group Sn 63 4.2 Frobenius-Young Modules 68 4.3 Specht Modules 73 4.4 General Case 76 4.5 Proof of Theorem 4.3.7 77 4.6 Representation Theory of An 79 Exercises 82 5 Symmetric Polynomials 83 5.1 Notation and Motivation 83 5.2 Several Bases for S (n) 84 5.3 Kostka Numbers & Determinantal Formulas 85 5.4 Results xλ 86 Exercises 91 6 Schur-Weyl Duality and the Relationship Between Representations of Sd and GLn (C) 92 6.1 Generalities 92 6.2 Schur-Weyl Duality 94 6.3 Characters of the Schur Modules 96 6.4 Schur Module Representations of SLn (C) 97 6.5 Representations of GLn (C) 98 Exercises 99 7 Structure Theory of Complex Semisimple Lie Algebras 101 7.1 Introduction to Semisimple Lie Algebras 101 7.2 The Exponential Map in Characteristic Zero 103 7.3 Structure of Semisimple Lie Algebras 103 7.4 Jordan Decomposition in Semisimple Lie Algebras 107 7.5 The Lie Algebra sln (C) 108 7.6 Cartan Subalgebras 109 7.7 Root Systems 110 7.8 Structure Theory of sln (C) 112 Exercises 113 8 Representation Theory of Complex Semisimple Lie Algebras 114 8.1 Representations of g 114 8.2 Weight Spaces 118 8.3 Finite Dimensional Modules 120 8.4 Fundamental Weights 123 8.5 Dimension and Character Formulas 124 8.6 Irreducible sln (C)-Modules 124 Exercises 124 9 Generalities on Algebraic Groups 126 9.1 Algebraic Groups and Their Lie Algebras 126 9.2 The Tangent Space 127 9.3 Jordan Decomposition in G 128 9.4 Variety Structure on G/H 132 9.5 The Flag Variety 136 9.6 Structure of Connected Solvable Groups 137 9.7 Borel Fixed Point Theorem 139 9.8 Variety of Borel Subgroups 143 Exercises 143 10 Structure Theory of Reductive Groups 145 10.1 Cartan Subgroups 145 10.2 The Weyl Group 146 10.3 Regular and Singular Tori 147 10.4 Semisimple Rank 1 149 10.5 One Parameter Subgroups 150 10.6 Reductive Groups 153 10.7 Almost Simple Groups 156 10.8 Schubert Varieties & Bruhat Decomposition 157 10.9 Standard Parabolic Subgroups 160 Exercises 160 11 Representation Theory of Semisimple Algebraic Groups 162 11.1 Weight Vectors 162 11.2 Geometric Realization of V(λ) 166 12 Geometry of the Grassmannian, Flag and their Schubert Varieties via Standard Monomial Theory 173 12.1 Grassmannian Variety 173 12.2 Gd,n Identified with a Homogeneous Space 176 12.3 Schubert Varieties 177 12.4 Standard Monomials 179 12.5 Equations Defining Schubert Varieties 181 12.6 Unions of Schubert Varieties 182 12.7 Vanishing Theorems 184 12.8 Results for F 187 Exercises 194 13 Singular Locus of a Schubert Variety in the Flag Variety SLn/B 195 13.1 Generalities 195 13.2 Singular Loci of Schubert Varieties 196 Exercises 200 14 Applications 201 14.1 Determinantal Varieties 201 14.2 Classical Invariant Theory 204 14.3 Toric Degeneration of a Schubert Variety 206 Exercises 209 15 Free Resolutions of Some Schubert Singularities. 210 15.1 One geometric technique 210 15.2 Schubert varieties and homogeneous bundles 213 15.3 Properties of Schubert desingularization 219 15.4 Free resolutions 224 15.5 Cohomology of Homogeneous Vector-Bundles 226 15.6 Examples 230 15.7 Further remarks 233 16 Levi Subgroup Actions on Schubert Varieties, and Some Geometric Consequences 237 16.1 Preliminaries 237 16.2 Decomposition Results 247 16.3 Multiplicity Consequences of the Decomposition 266 16.4 Sphericity Consequences of the Decomposition 282 16.5 Singularities and the L-action in Degree 1 283 Appendix A Chevalley Groups 290 A.1 A Basis for ε 290 A.2 Kostant's Z-form 291 A.3 Admissible Lattices 292 A.4 The Chevalley Groups 293 A.5 Simplicity of Groups of Adjoint Type 295 A.6 Chevalley Groups and Algebraic Groups 295 A.7 Generators, Relations of Universal Group 297 References 299 List of Symbols 305 Index 308 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. The book presents a discussion of complex semisimple Lie algebras and of semisimple algebraic groups. It also gives a detailed account of Grassmann varieties, flag varieties, and their Schubert subvarieties.