Alphabeta MathThe Degenerate Ariki Categorification Theorem
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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
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- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
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Algebraic Geometry
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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
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- KLR Algebras, Induction, and Quantum-Serre Categorification
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- Parabolic Category O and Fock-Space Bases
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Category Theory
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Commutative Algebra
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Complex Analysis
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Coxeter Groups
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Homological Algebra
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Lie Theory
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Linear Algebra
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Number Theory
Real Analysis
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- pi: the Equivalent Characterizations
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- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
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- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
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- Topology of ℝ
Representation Theory of Groups
Scheme Theory
- Affine Group Schemes, Hopf Algebras, and Rational Representations
- Affine Schemes and the Structure Sheaf
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- 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
- Integral Specht Modules and Modular Simple Modules
- Jucys–Murphy Elements and Seminormal Forms
- Kronecker Coefficients and Internal Products
- Specht Modules and the Irreducibles of the Symmetric Group
- The Branching Rule and the Young Graph
- The Hook Length Formula and Rsk Correspondence
- Young Diagrams Tableaux and Permutation Modules
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
The degenerate cyclotomic Hecke tower and its residue functors
Theorem
The degenerate cyclotomic Hecke G₀ is the dual highest weight module
Theorem
The degenerate cyclotomic Hecke projective K₀ is the highest weight module
Theorem
The degenerate Ariki categorification theorem
Remark
The degenerate and nondegenerate Ariki theorems have distinct parameter ranges
Theorem
Degenerate Specht decomposition numbers are canonical basis coefficients at one
Proposition
Sign duality and Ringel duality match the Fock-space involutions
Sources
- 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 and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008) 1-57 (complete 50-page arXiv article)
- Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians and finite W-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918 (complete 109-page arXiv memoir)
- Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, arXiv:0808.2032v4
- Jonathan Brundan and Alexander Kleshchev, The degenerate analogue of Ariki's categorification theorem, arXiv:0901.0057v1
- Brundan-Kleshchev, The degenerate analogue of Ariki's categorification theorem (complete 44-page arXiv article)
- Brundan-Kleshchev, Schur-Weyl duality for higher levels (complete 50-page arXiv article)