Alphabeta MathCyclotomic Hecke Quotients and Ariki–Koike PBW
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1 · Prerequisites
Abstract Algebra
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- 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
- 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
- Splitting Fields
- Tensor Products of Modules
- The Fundamental Theorem of Finite Abelian Groups
Categorification and Geometric Representation Theory
Coxeter Groups
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- 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
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
Set Theory
Symmetric Groups and Symmetric Functions
2 · Logical flowchart
3 · Items
Definition
The multiplicative cyclotomic Hecke algebra
Lemma
Cyclotomic Jucys–Murphy generators and their transport relations
Definition
Multipartitions, standard multitableaux, and generic contents
Lemma
Integral cyclotomic straightening and bounded PBW spanning
Theorem
Generic seminormal models for cyclotomic Hecke algebras
Lemma
Colored RSK and the multipartition square count
Theorem
The Ariki–Koike PBW basis and generic splitness
Lemma
The rank-one cyclotomic quotient has a monic basis
Sources
- Georges Neaime, Geodesic normal forms and Hecke algebras for the complex reflection groups G(de,e,n), arXiv:1810.12053v1
- Jun Hu and Andrew Mathas, Seminormal forms and cyclotomic quiver Hecke algebras of type A, arXiv:1304.0906v3
- David A. Craven, Groups, Geometries and Representation Theory, Spring Term 2013 lecture notes
- Georges Neaime, Geodesic normal forms and Hecke algebras for the complex reflection groups G(de,e,n)
- Andrew Mathas, Cyclotomic quiver Hecke algebras of type A (lecture notes)