Alphabeta MathUnequal-Parameter Hecke Cells and Representations
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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 over a Principal Ideal Domain and the Canonical Forms
- 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 Field of Fractions and Localisation
- The Fundamental Theorem of Finite Abelian Groups
Categorification and Geometric Representation Theory
Category Theory
Combinatorics
Commutative Algebra
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- Integral Extensions and Going Up
- Inverse Limits and Noetherian Completion
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Regular Local Rings and Homological Dimension
- Valuation Rings and Discrete Valuation Rings
Coxeter Groups
Group Theory
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- 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
2 · Logical flowchart
3 · Items
Definition
Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order
Definition
Canonical basic sets for split Hecke specializations
Definition
Schur elements of a split semisimple algebra with a symmetrizing trace
Remark
Lusztig P1–P15 are not universally proved cell properties
Lemma
Complete local modular systems and integral idempotent lifting
Definition
Weight functions and ordered group-algebra coefficients
Theorem
Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank
Lemma
Simple-generator weights extend uniquely to a weight function
Definition
The unequal-parameter Coxeter Hecke algebra
Lemma
The arbitrary-rank weighted Hecke basis and Bruhat-triangular bar
Theorem
Existence and uniqueness of the ordered-group unequal-parameter Kazhdan–Lusztig basis
Definition
Unequal-parameter left, right, and two-sided cells and cell quotients
Definition
Finite-Weyl Lusztig data and the precise basic-set hypotheses
Theorem
Weighted cell quotients are well-defined modules and bimodules
Lemma
The asymptotic algebra, filtered layers and the specialization homomorphism
Theorem
The four weighted cells of A1×A1 give a canonical basic set at odd residue characteristic
Lemma
Leading Schur coefficients and split good-characteristic asymptotic specializations
Lemma
Asymptotic induction identifies leading modular simples
Lemma
Filtered projectives prove canonical basic-set reciprocity
Theorem
Finite-Weyl canonical basic sets from Lusztig properties
Corollary
Weighted cell quotients and the leading finite-Weyl basic-set labels
Remark
The scope and proof status of the finite-Weyl basic-set theorem
Sources
- Anders Björner and Francesco Brenti, Combinatorics of Coxeter Groups, GTM 231 (2005)
- George Lusztig, Hecke Algebras with Unequal Parameters, revised arXiv text (2014)
- Meinolf Geck, Modular Representations of Hecke Algebras, EPFL lecture notes (2006)
- Meinolf Geck, Modular Representations of Hecke Algebras
- Meinolf Geck, Modular Representations of Hecke Algebras, sections 4–6
- George Lusztig, Hecke Algebras with Unequal Parameters, revised text, sections 18–21