Alphabeta MathHecke Symmetries and Generic Quantum Schur–Weyl Duality
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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
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- 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
2 · Logical flowchart
3 · Items
Definition
The ordered-basis Hecke operator and its normalization
Lemma
The ordered-basis operator is a Hecke symmetry
Theorem
The ordered-basis symmetry gives the normalized Hecke tensor action
Definition
The candidate antipode for the type-A quantum algebra
Lemma
The type-A quantum presentation admits its displayed antipode
Lemma
The quantum and Hecke tensor actions commute
Lemma
Classical tensor centralizer from polarization and Newton identities
Lemma
Finite semisimple matrix density and the double commutant over Q
Lemma
Finite matrix rank bounds under regular specialization
Lemma
Classical commutants bound the two generic tensor images
Theorem
Generic quantum Schur–Weyl mutual centralizers over Q(q)
Sources
- Alexei Davydov and Alexander Molev, A categorical approach to classical and quantum Schur–Weyl duality
- Meinolf Geck, Modular Representations of Hecke Algebras
- Pavel Etingof and Mykola Semenyakin, A brief introduction to quantum groups
- Richard Dipper, Stephen Doty and Friederike Stoll, Quantized mixed tensor space and Schur–Weyl duality
- Pavel Etingof et al., Introduction to representation theory