Alphabeta MathAriki Categorification and Graded Decomposition Numbers
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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
- Free Products and Amalgamation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Induced Representations, Frobenius Reciprocity and Applications
- 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
- Sylow's Theorems, p-Groups and Nilpotent Groups
- 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
- Affine Perverse Vanishing And Universal Hyperplanes
- Algebraic Differentials Separability and Smooth Local Presentations
- Analytic Constructible Sheaf Foundations
- Chow Groups, Intersection Products, and Grothendieck-Riemann-Roch
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Complex Algebraic Stratified Topology
- Constructible Incidence And Cochain Models
- Dimension Constructible Images and Dimensions of Fibres
- Geometric Fixed Parts And Gauss Manin Semisimplicity
- Higher-Dimensional Resolution of Singularities
- Logarithmic Hodge Complexes And Smooth Open Weights
- Mixed Hodge Structures And Strictness
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties
- Point Blowup Resolution on Arbitrary Regular Surfaces
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
- Projective Perverse Hard Lefschetz And Decomposition
- Proper Hyperresolutions And Mixed Hodge Restriction Images
- Smooth-Projective Serre Duality and Flag-Variety Line Bundles
- Unipotent and Solvable Groups and Borel Fixed Points
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
Algebraic Topology
- Chern and Pontryagin Classes by Splitting and Complexification
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Higher Homotopy Groups and Cofiber Sequences
- Homology Axioms Degree and Classical Applications
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Leray–Hirsch, the Thom Isomorphism, and Gysin Sequences
- Local Coefficients, Twisted Homology, and Duality
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- Orientations Poincare Lefschetz and Alexander Duality
- Relative Homology Excision and Mayer Vietoris
- Simple Homotopy, Whitehead Groups, and Torsion
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
- Stiefel Whitney and Euler Classes by Universal Constructions
- The Serre Spectral Sequence and Applications
- Topological Vector Bundles and Grassmannian Classification
Braid Groups
Categorification and Geometric Representation Theory
- Arakawa–Suzuki Functors and Higher-Level Schur–Weyl Duality
- Balanced Triples and Global Crystal Bases
- Cyclotomic Hecke and Quiver Hecke Algebras
- Cyclotomic KLR Algebras and Highest-Weight Categorification
- Divided Powers, Integral Forms, and Quantum Highest Weights
- Equivariant Borel–Moore Correspondence Convolution and Constructible Ext
- Kashiwara Operators and Crystal Lattices
- Kazhdan–Lusztig Bases, Polynomials, and Cells
- KLR Algebras, Induction, and Quantum-Serre Categorification
- Level-One Fock Space and Residue Crystals
- Ordinary Hecke Specht Polytabloids and Signed Murphy Comparison
- Proper Cohomological Descent And Constructible Hypercovers
- Quantized Enveloping Algebras and Quantum Serre Relations
- Quantum Affine Type A and Level-One Fock Space
- Tensor Coherence and Algebraic Descent
- Tensor-Product Crystals and Highest-Weight Crystal Uniqueness
- Type-A Hecke Specht Modules and Cellular Bases
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
- Tensor and Fusion Categories
- 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
- Henselian Rings and Equicharacteristic Cohen Structure
- Homogeneous Resultants and Projective Intersection Length
- Integral Extensions and Going Up
- Inverse Limits and Noetherian Completion
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Normalization Finiteness for Affine Domains
- 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
- Analyticity of Holomorphic Functions; Liouville and Morera
- Compact Kahler Hodge Foundations
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Divisors, Riemann--Roch, and Duality
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Holomorphic Functions of Several Complex Variables
- Isolated Singularities and Laurent Series
- Projective Hodge Polarization Foundations
- Riemann Surfaces, Branched Maps, and Differentials
- The Argument Principle and Rouché's Theorem
- The Dolbeault Complex and Integral Solutions
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Residue Theorem and the Evaluation of Real Integrals
- The Riemann Sphere and Möbius Transformations
- The Winding Number and the Global Cauchy Theorem
Coxeter Groups
Differential Geometry
- Chern–Weil Theory and Characteristic Forms
- Compact Laplace-type operators and harmonic representatives
- Connections Levi Civita and Parallel Transport
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Measurable Densities and Radon Volume on Manifolds
- Rank Theorems and Embedded Submanifolds
- Riemann Curvature and Riemannian Submanifolds
- Riemannian Metrics Length Distance and Volume
- Sard Theorem and Transversality
- Singular Cochains Mayer Vietoris and Smooth Singular Comparison
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The De Rham Complex Homotopy and Mayer Vietoris
- The de Rham Theorem and Degree
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
- Whitney Embedding Tubular Neighbourhoods and Approximation
Differential Topology
- Exotic Smooth Structures and Milnor Spheres
- Gradient Like Vector Fields and Morse Trajectories
- Handle Decompositions Duality and Rearrangement
- Intersection Pairings Self Intersection and Euler Classes
- Morse Critical Points Hessians and Indices
- Morse Functions Critical Values and Genericity
- Reeb Stability and Global Foliation Constructions
- Relative smooth triangulation and bundle fitting
- Sublevel Deformation and the Handle Attachment Theorem
Functional Analysis
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- Complex Norms and Realification
- Distributions Test Functions and Differentiation
- Dual Spaces Adjoint Operators and Annihilators
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Orthonormal Bases, Parseval and Fourier Series
- Reflexivity and Eberlein Smulian
- Schwartz Space and the Plancherel Theorem
- The Analytic Hahn Banach Theorem
Group Theory
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Delta Functors and Universality
- Derived Categories
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Graded Bimodules and Tensor Functors
- Grothendieck Groups and Graded Cartan Pairings
- Group Cohomology as a Derived Functor
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Lefschetz Splitting In Bounded Triangulated Categories
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Spectral Sequences
- Tor Flatness and Global Dimension
- Triangulated Categories
- Universal Coefficients and Kunneth Theorems
- Yoneda Extensions and Homological Dimension
Lie Theory
- Finite Weyl Invariants, Bruhat Order, and Kostant Harmonics
- Harish Chandra Isomorphism Casimir and Central Characters
- 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
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
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Absolute Continuity and the Sharp Fundamental Theorem of Calculus
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Differentiation of Monotone Functions and the Vitali Covering Theorem
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Radon Measures and the Riesz Markov Kakutani Theorem
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Duality of Lᵖ and L^q
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
Number Theory
Partial Differential Equations
- Euclidean Surface Measure, Divergence, and Green Identities
- Fredholm Elliptic Problems and the Elliptic Spectrum
- Harmonic Functions and Mean Values in Rn
- Interior and Boundary Sobolev Elliptic Regularity
- Lax--Milgram and Weak Elliptic Solutions
- Poisson Problems and Interior Harmonic Estimates
- Rellich Kondrachov and Sobolev Compactness
- Smooth Approximation and Sobolev Extension
- Sobolev Poincare and Morrey Inequalities
- Sobolev Traces and Zero Boundary Values
- Weak Derivatives and Sobolev Spaces
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- 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
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Improper and Parameter-Dependent Multiple Integrals
- 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 Divergence Theorem and Classical Stokes
- 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
- Volumes of Elementary Solids and Solids of Revolution
Representation Theory of Groups
- Blocks Defect Groups and the Brauer Homomorphism
- Bruhat Decomposition and Flags over Finite Fields
- Equivariant Constructible Perverse Foundations
- Galois Orbits and Descent of Simple Finite-Group Modules
- Principal Series Representations of GL N over a Finite Field
- Quiver Full Flag Projective Semisimplicity
- Signed Quiver Flag Convolution And Klr Ext Realization
Scheme Theory
- Affine Group Schemes, Hopf Algebras, and Rational Representations
- Affine Schemes and the Structure Sheaf
- Algebraic Group Actions, Orbits, Stabilizers, and Controlled Quotients
- Algebraic Spaces, Stacks, and Derived Algebraic Geometry Foundations
- Analytic Derived Six Operations And Duality
- Blowups, Exceptional Divisors, and Strict Transforms
- Cartier and Weil Divisors Line Bundles and Picard Groups
- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
- Compact Support Sheaf Calculus
- 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
- Presheaves Sheaves Stalks and Sheafification
- Proj Projective Schemes Twisting Sheaves and Ampleness
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Residues Serre Duality for Curves and the Full Riemann Roch Theorem
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Cohomology Cech Cohomology and Comparison
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
- Smooth Proper Curves Divisors Genus and Ramification
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
- Cores, Quotients, and Blocks of Symmetric Groups
- Integral Specht Modules and Modular Simple Modules
- Jucys–Murphy Elements and Seminormal Forms
- Modular Branching and the Good-Node Crystal
- Rim Hooks and the Murnaghan–Nakayama Rule
- Specht Modules and the Irreducibles of the Symmetric Group
- Symmetric Functions, the Hall Inner Product, and Schur Bases
- The Branching Rule and the Young Graph
- The Hook Length Formula and Rsk Correspondence
- Young Diagrams Tableaux and Permutation Modules
Topology
- Classification of Covering Spaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- 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 Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
The domination order on tableaux, Garnir tableaux and bricks
Lemma
Generic nondegenerate local inverse tables on integer colors
Lemma
An integer cyclic presentation descends the simple-head generator line
Lemma
Integer-color degenerate Hecke–KLR local inverse tables
Lemma
Generic cyclotomic type-A support is bounded by rank
Lemma
Generic cyclotomic local dictionaries glue to a tower
Lemma
Extremal word characters detect normalized cyclotomic simple labels
Lemma
Standard Fock vectors expand in the upper global basis
Definition
Graded module and projective Grothendieck groups of the KLR and Hecke towers
Definition
The nondegenerate root-of-unity level-one Hecke regime
Definition
The graded Cartan pairing between projective and module classes
Definition
The graded Specht module by homogeneous standard tableaux
Definition
The integral form of the cyclotomic Hecke-KLR block
Lemma
Degree and codegree of standard tableaux
Lemma
A graded algebra isomorphism transports the graded module category, its Grothendieck groups and its dualities
Proposition
The QG-8 categorification transports across the blockwise Hecke-KLR bridge
Proposition
Shifted degrees and Garnir exchanges for standard tableaux
Theorem
The graded Specht module has a homogeneous tableau basis
Lemma
Rational Garnir annihilator lines present the specified Specht generator
Theorem
Graded Specht restriction has the removable-node filtration
Theorem
Graded Specht classes realize the standard Fock vectors
Corollary
The restricted graded Specht classes form a basis of the module Grothendieck group
Theorem
Ariki's canonical basis theorem in characteristic zero
Theorem
Graded decomposition numbers equal canonical-basis coefficients in characteristic zero
Theorem
Graded Specht heads classify the graded simple modules
Lemma
Reduction of characteristic-zero simple modules modulo p
Lemma
Positive-characteristic Specht head and cyclotomic crystal labels agree
Definition
The graded adjustment polynomials in positive characteristic
Theorem
Positive-characteristic decomposition polynomials factor through adjustment
Remark
Canonical bases do not solve arbitrary positive-characteristic decomposition numbers
Sources
- Jonathan Brundan, Alexander Kleshchev and Weiqiang Wang, Graded Specht modules, arXiv:0901.0218v3, J. reine angew. Math. 655 (2011) 61-87 (complete 23-page article)
- Jun Hu and Andrew Mathas, Graded induction for Specht modules, arXiv:1008.1462v1 (complete 19-page article)
- Brundan and Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, full arXiv:0808.2032v4
- Brundan–Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras, arXiv:0808.2032v4
- Brundan and Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
- Jonathan Brundan and Alexander Kleshchev, Graded decomposition numbers for cyclotomic Hecke algebras, arXiv:0901.4450v3
- Jonathan Brundan and Alexander Kleshchev, Graded decomposition numbers for cyclotomic Hecke algebras, arXiv:0901.4450v3, Adv. Math. 222 (2009) 1883-1942 (complete 57-page article)
- Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, arXiv:0909.4844v1, Bull. Amer. Math. Soc. 47 (2010) 419-481 (complete 66-page survey)
- Susumu Ariki, On the decomposition numbers of the Hecke algebra of G(m,1,n), J. Math. Kyoto Univ. 36 (1996) 789-808
- Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete 79-page prepublication manuscript
- Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, arXiv:0808.2032v4, Invent. Math. 178 (2009) 451-484 (complete 32-page article)
- Brundan–Kleshchev–Wang, Graded Specht modules, arXiv0901.0218v3
- Kleshchev–Mathas–Ram, Universal graded Specht modules for cyclotomic Hecke algebras, arXiv1102.3519v1
- Matthew Fayers, Weight two blocks of Iwahori–Hecke algebras in characteristic two