Alphabeta MathParabolic Category O and Fock-Space Bases
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Characters and the Orthogonality Relations
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Averaging and Character-Theory Prerequisites
- Finite Fields and Cyclotomic Extensions
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Maschke's Theorem, Complete Reducibility and the Structure of k[G]
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Solvability by Radicals and Kummer Theory
- Splitting Fields
- Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Dimension Constructible Images and Dimensions of Fibres
- Groups of Multiplicative Type and Arithmetic Tori
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
- Unipotent and Solvable Groups and Borel Fixed Points
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
Braid Groups
Categorification and Geometric Representation Theory
- Balanced Triples and Global Crystal Bases
- Divided Powers, Integral Forms, and Quantum Highest Weights
- Kashiwara Operators and Crystal Lattices
- Kazhdan–Lusztig Bases, Polynomials, and Cells
- Projective Functors and the Soergel Functor in Category O
- Quantized Enveloping Algebras and Quantum Serre Relations
- Singular and Parabolic Kazhdan–Lusztig Theory
- Soergel Intersection Forms and Hodge Theory
- Tensor-Product Crystals and Highest-Weight Crystal Uniqueness
- The Kazhdan–Lusztig Character and Multiplicity Theorem
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Braided and Symmetric Monoidal Categories
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
Commutative Algebra
- Algebraic Zariski Main for Quasi-Finite Morphisms
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Dedekind Domains and Ideal Classes
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- Homogeneous Resultants and Projective Intersection Length
- Integral Extensions and Going Up
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Regular Local Rings and Homological Dimension
- Valuation Rings and Discrete Valuation Rings
- Zariski Topology on Prime Spectra
Complex Analysis
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Holomorphic Functions of Several Complex Variables
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Coxeter Groups
- Canonical Roots, Signs, and Faithful Reflections
- Conjugacy in Sₙ, Generation, and the Simplicity of Aₙ
- Coxeter Presentations, Exchange, and Reduced Word Theorems
- Parabolic Subgroups and Double Coset Geometry
- Permutation Statistics, Inversions and Eulerian Numbers
- Real Forms and Reflection Geometry
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
Differential Geometry
- Distributions Integral Manifolds and the Frobenius Theorem
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Vector Fields Flows and Lie Derivatives
Functional Analysis
Group Theory
Homological Algebra
- Bounded Bimodule Complexes and Derived Tensor
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Delta Functors and Universality
- Derived Categories
- Derived Functors
- Ext and Balanced Resolutions
- Graded Bimodules and Tensor Functors
- Grothendieck Groups and Graded Cartan Pairings
- Homological Gaussian Elimination
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Tor Flatness and Global Dimension
- Triangulated Categories
- Universal Coefficients and Kunneth Theorems
- Yoneda Extensions and Homological Dimension
Lie Theory
- Cartan Subalgebras and Root Space Decompositions
- Category O Finiteness Duality and Blocks
- Finite Weyl Invariants, Bruhat Order, and Kostant Harmonics
- Harish Chandra Isomorphism Casimir and Central Characters
- Highest Weight Theory for Complex Semisimple Lie Algebras
- Homomorphisms Between Verma Modules and Linkage
- Integrable Highest Weight Kac Moody Modules
- Kac Moody Algebras from Generalized Cartan Matrices
- Lie Algebra Representations, Enveloping Algebras, and PBW
- Lie Groups, Invariant Fields, and the Exponential Map
- Lie Subgroups, Actions, and Homogeneous Spaces
- Projectives Standard Filtrations and Bgg Reciprocity
- Root Systems, Dynkin Diagrams, and the Cartan-Killing Classification
- Semisimple Lie Algebras, Cohomology, and Levi Theory
- Solvable and Nilpotent Lie Algebras
- The BGG Resolution
- The Weyl Kac Character Formula
- Verma Modules and Shapovalov Forms
- Weyl Character and Multiplicity Formulas
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Representation Theory of Groups
Scheme Theory
- Affine Group Schemes, Hopf Algebras, and Rational Representations
- Affine Schemes and the Structure Sheaf
- Algebraic Group Actions, Orbits, Stabilizers, and Controlled Quotients
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale Morphisms
- Group Schemes of Finite Type over a Field
- Kahler Differentials Conormal Sequences and Infinitesimal Lifting
- Lie Algebras and Infinitesimal Group Schemes
- Presheaves Sheaves Stalks and Sheafification
- Proj Projective Schemes Twisting Sheaves and Ampleness
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Symmetric Groups and Symmetric Functions
Topology
- Applications of the Fundamental Group
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
2 · Logical flowchart
3 · Items
Definition
Standard- and costandard-filtered objects and tilting objects
Definition
The Fock-space convention dictionary for finite type-A quantum modules
Definition
The higher-level wedge-tensor Fock module
Lemma
Parabolic semi-regular duality and the reverse-column label map
Theorem
The column monomial basis of the higher-level Fock module
Theorem
The tensor-product bar on the Fock module and its truncation compatibility
Definition
The dual-canonical, canonical and quasi-canonical Fock bases
Lemma
Finite type-A tensor Hecke bars and complementary coefficient charts
Lemma
Full finite type-A tensor source upper charts and string leading terms
Lemma
Type-A tableau central summands are single linkage blocks
Theorem
Standard tableau monomials form a basis of the highest weight Fock submodule
Definition
The integral parabolic category-O blocks for higher-level Schur-Weyl duality
Definition
The residue translation functors on the sum of parabolic blocks
Theorem
Parabolic Kazhdan-Lusztig polynomials give the dual-canonical transition matrix
Lemma
Residue Casimir operators and local affine-Hecke actions
Theorem
The highest weight Fock submodule carries the quantum-group global bases
Lemma
The local affine-Hecke socle argument for residue translations
Theorem
The residue translation functors give the integrable Fock-space action
Theorem
Indecomposable tilting objects exist and are classified in parabolic category O
Theorem
The parabolic category-O Grothendieck group is the integral Fock module
Definition
The Ringel dual of a finite highest weight ideal
Lemma
The Fock pairing matches the projective-simple and standard-costandard pairings
Theorem
Type-A parabolic category-O Ringel duality via the opposite parabolic
Theorem
Simples, projectives and tiltings map to the three distinguished Fock bases
Sources
- Volodymyr Mazorchuk, Lectures on Algebraic Categorification, QGM Master Class Series, European Mathematical Society 2012 (complete 77-page arXiv lecture-note set)
- Jonathan Brundan and Alexander Kleshchev, The degenerate analogue of Ariki's categorification theorem, Math. Z. 266 (2010) 877-919 (complete 44-page arXiv article)
- Jonathan Brundan, Dual canonical bases and Kazhdan-Lusztig polynomials, J. Algebra 306 (2006) 17-46 (complete 29-page arXiv v2 article)
- Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008) 1-57 (complete 50-page arXiv article)
- Wolfgang Soergel, Character formulas for tilting modules over Kac-Moody algebras (complete 22-page author English preprint; some later additions absent)
- Jonathan Brundan, Dual canonical bases and Kazhdan-Lusztig polynomials
- Masaki Kashiwara, Global crystal bases of quantum groups
- Shun-Jen Cheng, Weiqiang Wang and R.B. Zhang, A super duality and Kazhdan-Lusztig polynomials (complete45-page arXiv0409016)
- Jonathan Brundan, Dual canonical bases and Kazhdan-Lusztig polynomials (complete29-page arXiv0509700v2)
- Joseph Chuang and Raphael Rouquier, Derived equivalences for symmetric groups and sl2-categorification (complete45-page arXiv0407205)
- Masaki Kashiwara, Global crystal bases of quantum groups, Duke Math. J.69 (1993)455–485 (complete33-page author PDF)
- Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians and finite W-algebras (complete109-page arXiv0508003)
- Jonathan Brundan, Tilting modules for Lie superalgebras, Comm. Algebra 32 (2004) 2251-2268 (complete arXiv article)
- Henning Haahr Andersen and Catharina Stroppel, Twisting functors on O, Represent. Theory 7 (2003) 681-699 (author-hosted complete article, University of Bonn)