Alphabeta MathType A Affine Hecke Algebras and Bernstein 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
2 · Logical flowchart
3 · Items
Lemma
The finite type-A Hecke basis over the multiplicative parameter ring
Definition
The type-A weight lattice, Laurent ring, and divided differences
Definition
The rational skew group algebra for type A
Definition
The multiplicative Bernstein presentation of the affine Hecke algebra of type A
Lemma
Demazure–Lusztig operators satisfy the affine type-A relations
Theorem
Bernstein PBW basis for the affine Hecke algebra of type A
Theorem
The center of the universal affine Hecke algebra of type A
Sources
- George Lusztig, Lectures on Hecke Algebras with Unequal Parameters, MIT lecture notes, arXiv:math/0108172
- Anders Björner and Francesco Brenti, Combinatorics of Coxeter Groups, Springer GTM 231 (2005), complete author-hosted book
- I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki 797, Astérisque 237 (1996), pp. 189–207
- Alexander A. Kirillov, Jr., Lectures on the affine Hecke algebras and Macdonald conjectures (Harvard lectures, 1994; arXiv:math/9501219v1)
- Eric Opdam and Maarten Solleveld, Homological algebra for affine Hecke algebras (2007)