دانلود کتاب Introduction to Soergel Bimodules

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Preface How to Read This Book About This Book and Acknowledgements Leitfaden Contents Contributors Part I The Classical Theory of Soergel Bimodules 1 How to Think About Coxeter Groups 1.1 Coxeter Systems and Examples 1.1.1 Definition of a Coxeter System 1.1.2 Example: Type A 1.1.3 Example: Type B 1.1.4 Example: Type D 1.1.5 Example: Dihedral Groups 1.1.6 Coxeter Groups and Reflections 1.1.7 The Geometric Representation and the Classification of Finite Coxeter Groups 1.1.8 Crystallographic Coxeter Systems 1.2 Coxeter Group Fundamentals 1.2.1 The Length Function 1.2.2 The Descent Set 1.2.3 The Exchange Condition 1.2.4 The Longest Element 1.2.5 Matsumoto's Theorem 1.2.6 Bruhat Order 1.2.7 Additional Exercises 2 Reflection Groups and Coxeter Groups 2.1 Reflections and Affine Reflections 2.2 Affine Reflection Groups 2.3 Affine Reflection Groups are Coxeter Groups 2.4 Expressions and Strolls 2.5 Classification of Affine Reflection Groups 2.6 The Coxeter Complex 3 The Hecke Algebra and Kazhdan–Lusztig Polynomials 3.1 The Hecke Algebra 3.1.1 The Standard Basis 3.1.2 Inversion 3.2 The Kazhdan–Lusztig Basis 3.2.1 The Standard Form on H 3.3 Existence of the Kazhdan–Lusztig Basis 3.3.1 A Motivating Example 3.3.2 Construction of the Kazhdan–Lusztig Basis 3.3.3 The Kazhdan–Lusztig Presentation 3.3.4 Deodhar's Formula 4 Soergel Bimodules 4.1 Gradings 4.2 Polynomials 4.2.1 Invariant Polynomials 4.3 Demazure Operators 4.4 Bimodules and Tensor Products 4.5 Bott–Samelson Bimodules 4.6 Soergel Bimodules 4.7 Examples of Soergel Bimodules 4.8 A First Glimpse of Categorification 5 The ``Classical'' Theory of Soergel Bimodules 5.1 Twisted Actions 5.2 Standard Bimodules 5.3 Soergel Bimodules and Standard Filtrations 5.4 Localization 5.5 Soergel's Categorification Theorem 5.6 A Technical Wrinkle 5.7 Realizations of a Coxeter System 5.8 More Technicalities 6 Sheaves on Moment Graphs 6.1 Roots in the Geometric Representation 6.2 Bruhat Graphs 6.3 Moment Graphs 6.4 Sheaves on Moment Graphs 6.5 The Braden–MacPherson Algorithm 6.6 Stalks and Standard Bimodules 6.7 A Functor to Sheaves on Moment Graphs 6.8 Soergel's Conjecture and the Braden–MacPherson Algorithm Part II Diagrammatic Hecke Category 7 How to Draw Monoidal Categories 7.1 Linear Diagrams for Categories 7.2 Planar Diagrams for 2-Categories 7.3 Drawing Monoidal Categories 7.4 The Temperley–Lieb Category 7.5 More About Isotopy 8 Frobenius Extensions and the One-Color Calculus 8.1 Frobenius Structures 8.1.1 Frobenius Algebra Objects 8.1.2 Diagrammatics for Frobenius Algebra Objects 8.1.3 Playing with Isotopy 8.1.4 Frobenius Extensions 8.2 A Tale of One Color 8.2.1 Frobenius Structure 8.2.2 Additional Generators and Relations 8.2.3 The Moral of the Tale 8.2.4 A Direct Sum Decomposition, Diagrammatically 9 The Dihedral Cathedral 9.1 A Tale of Two Colors 9.2 The Temperley–Lieb 2-Category 9.3 Jones–Wenzl Projectors 9.4 Two-color Relations 10 Generators and Relations for Bott–Samelson Bimodules and the Double Leaves Basis 10.1 Why Present BSBim? 10.2 Generators and Relations 10.2.1 A Diagrammatic Reminder 10.2.2 An Isotopy Presentation of HBS 10.2.3 Examples and Exercises 10.2.4 A Presentation of HBS 10.2.5 The Functor to Bimodules 10.2.6 General Realizations 10.3 Rex Moves and the 3-color Relations 10.4 Light Leaves and Double Leaves 10.4.1 Overview 10.4.2 The Algorithm 10.4.3 Diagrammatics and Bimodules 10.4.4 Light Leaves and Localization 11 The Soergel Categorification Theorem 11.1 Introduction 11.2 Prelude: From HBS to H 11.2.1 Graded Categories 11.2.2 Additive Closure 11.2.3 Karoubian Closure 11.2.4 Karoubi Envelopes Are Krull–Schmidt 11.2.5 Diagrammatics and Karoubi Envelopes 11.3 Grothendieck Groups of Object-Adapted Cellular Categories 11.3.1 Object-Adapted Cellular Categories 11.3.2 First Properties 11.3.3 The Main Example 11.3.4 Classifying Indecomposables in Object-Adapted Cellular Categories Appendix 1: Krull–Schmidt Categories Categories with Unique Decompositions Krull–Schmidt Categories The Karoubian Property Semiperfect Rings The Split Grothendieck Group of a Category with Shift Functor Appendix 2: Composition Forms, Cellular Forms, and Local Intersection Forms 12 How to Draw Soergel Bimodules 12.1 The 01-Basis 12.2 Commutative Ring Structure on a Bott–Samelson Bimodule 12.3 Trace and the Global Intersection Form 12.4 Bott–Samelson Bimodules and the Light Leaves Basis 12.5 Light Leaves Basis and the Standard Filtration on Bott–Samelson Bimodules Appendix 1: A Crucial Positivity Result Part III Historical Context: Category Oand the Kazhdan–Lusztig Conjectures 13 Category O and the Kazhdan–Lusztig Conjectures 13.1 Introduction 13.2 The Verma Problem 13.2.1 Verma Modules 13.2.2 Category O and Its Mysteries 13.3 The Kazhdan–Lusztig Conjectures 13.3.1 The Multiplicity Conjecture 13.3.2 Positivity and Schubert Varieties 13.4 Two Proofs of the Kazhdan–Lusztig Conjecture 14 Lightning Introduction to Category O 14.1 Lie Algebra Basics 14.2 Category O 14.3 Duality in O 14.3.1 Standard Filtrations and BGG Reciprocity 14.4 Blocks of Category O 14.5 Example: g = sl3(C) 15 Soergel's V Functor and the Kazhdan–Lusztig Conjecture 15.1 Brief Reminder on Category O 15.2 Translation Functors 15.2.1 Tensor Products 15.2.2 Definition of Translation Functors and First Properties 15.2.3 Effect on Verma Modules 15.2.4 Wall-Crossing Functors 15.2.5 Effect on Projective Modules 15.3 Soergel Modules 15.4 Soergel's V Functor 15.5 Soergel's Approach to the Kazhdan–Lusztig Conjecture 16 Lightning Introduction to Perverse Sheaves 16.1 Motivation 16.2 Stratified Spaces and Examples 16.2.1 Stratified Resolutions and Schubert Varieties 16.2.2 Constructible Sheaves and Pushforwards 16.2.3 Perverse Sheaves 16.2.4 The Decomposition Theorem 16.2.5 Connection to the Hecke Algebra Part IV The Hodge Theory of Soergel Bimodules 17 Hodge Theory and Lefschetz Linear Algebra 17.1 Introduction 17.2 Hard Lefschetz 17.3 Hodge–Riemann Bilinear Relations 17.4 Lefschetz Lemmas 18 The Hodge Theory of Soergel Bimodules 18.1 Introduction 18.2 Overview and Preliminaries 18.2.1 The Conjectures of Soergel and Kazhdan–Lusztig 18.2.2 Duality and Invariant Forms 18.2.2.1 Morphisms Between Soergel Bimodules 18.2.2.2 Invariant Forms 18.2.2.3 Invariant Forms on Soergel Bimodules 18.2.2.4 Lefschetz Forms and Positivity 18.2.2.5 Key Statements in the Induction 18.2.2.6 Induced Forms 18.2.3 The Main Theorem 18.3 Outline of the Proof 18.3.1 Step 1 18.3.1.1 Deforming the Lefschetz Operator 18.3.1.2 Flowchart for Step 1 18.3.2 Step 2 18.3.2.1 The Local Intersection Form 18.3.2.2 HR(x,s) Versus HR(x s) 18.3.2.3 Flowchart for Step 2 18.4 The Weak Lefschetz Problem 18.5 From zeta = 0 to zeta >> 0 18.6 From Local to Global Intersection Forms 18.7 Hodge Theory of Matroids 19 Rouquier Complexes and Homological Algebra 19.1 Motivation 19.2 Some Homological Algebra 19.2.1 Complexes and Homotopies 19.2.2 Gaussian Elimination and Minimal Complexes 19.2.3 Grothendieck Groups 19.3 Rouquier Complexes and Categorification of the Braid Group 19.4 Cohomology of Rouquier Complexes 19.5 Perversity 19.6 The Diagonal Miracle Appendix: More Homological Algebra Triangulated Structure Triangulated Grothendieck Groups Perverse t-Structure 20 Proof of the Hard Lefschetz Theorem 20.1 Introduction 20.2 Preliminaries 20.3 The Hodge–Riemann Relations for the Rouquier Complex 20.4 Positivity of Breaking 20.5 Sketch of the Proof of Hard Lefschetz Appendix: Some Historical Context and Geometric Intuition for the Proof of Soergel's Conjecture The Kazhdan–Lusztig Conjecture for Weyl Groups and the Decomposition Theorem What We Need I: Hard Lefschetz in a Family What We Need II: A Substitute for Hyperplane Sections Part V Special Topics 21 Connections to Link Invariants 21.1 Temperley–Lieb Algebra 21.2 Schur–Weyl Duality 21.3 Trace and Link Invariants 21.4 Quantum Groups and Link Invariants 21.5 Ocneanu Trace and HOMFLYPT Polynomial 21.6 Categorification of Braids and of the HOMFLYPT Invariant 21.6.1 Rouquier Complexes 21.6.2 Hochschild Homology 21.6.3 Categorifying the Standard Trace 22 Cells and Representations of the Hecke Algebra in Type A 22.1 Cells 22.1.1 Cells for a Monoidal Category 22.1.2 Cell Module Categories 22.1.3 Cells for a Based Algebra 22.2 Cells in Type A 22.2.1 Young Diagrams and Tableaux 22.2.2 The Robinson–Schensted Correspondence 22.2.3 Cells in Type A 22.2.4 The k-row Quotient of the Hecke Algebra 22.3 Representations of the Hecke Algebra in Type A 23 Categorical Diagonalization 23.1 Classical Linear Algebra 23.2 Categorified Linear Algebra 23.2.1 Eigenobjects 23.2.2 Prediagonalizability 23.2.3 Twisted Complexes 23.2.4 Diagonalizability 23.2.5 Smallness 23.2.6 Lagrange Interpolation 23.3 A Toy Example 23.4 Diagonalizing the Full Twist 23.4.1 Type A1 23.4.2 Type A 23.4.3 The General Case 24 Singular Soergel Bimodules and Their Diagrammatics 24.1 The Classical Theory of Singular Soergel Bimodules 24.1.1 Bimodules and Functors 24.1.2 Singular Soergel Bimodules 24.1.3 Categorification Theorems 24.2 One-Color Singular Diagrammatics 24.2.1 Diagrammatics for a Frobenius Extension 24.2.2 Relationship to the One-Color Soergel Calculus 24.2.3 Relationship with the Temperley–Lieb 2-Category 24.3 Singular Soergel Diagrammatics in General 24.3.1 The Upgraded Chevalley Theorem, Part I 24.3.2 The Upgraded Chevalley Theorem, Part II 24.3.3 Diagrammatics for a Cube of Frobenius Extensions 24.3.4 The Jones–Wenzl Relation 24.3.5 Additional Relations, Applications, and Future Work 24.3.6 Other Realizations 25 Koszul Duality I 25.1 Introduction 25.1.1 Morita Theory 25.2 dg-Algebras 25.3 dg-Morita Theory 25.4 Koszul Duality for Polynomial Rings 25.5 Review of the Kazhdan–Lusztig Conjecture 25.6 Evidence of Koszul Duality in Category O 26 Koszul Duality II 26.1 Introduction 26.2 Graded Category O0 26.2.1 Desiderata 26.2.2 Motivation from Soergel's V Functor 26.2.3 Definition of Soergel Category O0 26.2.4 Example: Soergel O0 in Type A1 26.3 Homological Properties of Soergel O0 26.3.1 Highest Weight Structure 26.3.2 Tilting Objects and the Realization Functor 26.3.3 Ringel Duality 26.4 Koszul Duality 26.4.1 Statement 26.4.2 Monodromy Action 26.4.3 Wall-Crossing Functors 26.4.4 Outline of the Proof of Theorem 26.26 26.5 Some Odds and Ends 27 The p-Canonical Basis 27.1 Introduction 27.2 Definition of the p-Canonical Basis 27.3 Computing the p-Canonical Basis 27.4 Geometric Incarnation of the Hecke Category 27.4.1 Parity Complexes on Flag Varieties 27.4.2 Parity Complexes and the Hecke Category 27.5 Modular Representation Theory of Reductive Groups 27.5.1 Soergel's Modular Category O 27.5.2 The Riche–Williamson Conjecture 27.5.3 This Is Not the End References Index