Alphabeta MathGeneric Coxeter Hecke Algebras and the Standard Basis
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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
2 · Logical flowchart
3 · Items
Definition
Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy
Lemma
The commuting left and right length operators and their Hecke relations
Lemma
Reduced-word independence of T_w and the length-multiplication rules
Theorem
The standard basis of the generic Hecke algebra and base change
Lemma
The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization
Sources
- George Lusztig, Lectures on Hecke Algebras with Unequal Parameters (MIT Fall 1999 lecture notes, arXiv:math/0108172v1)
- Meinolf Geck, Modular Representations of Hecke Algebras (EPFL course notes, arXiv:math/0511548v2)
- Anders Bjorner and Francesco Brenti, Combinatorics of Coxeter Groups, Springer GTM 231 (2005), complete book PDF