Alphabeta MathHecke Base Change, Semisimplicity, and Deformation
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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 Field of Fractions and Localisation
- 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
Finite Hecke specialization and its regular-trace determinant
Lemma
The one-parameter formal-series ring is complete, separated, and has residue field k
Lemma
Finite direct summands over a formal-series ring are free with rank read from reduction
Lemma
Orthogonal idempotents lift over a complete formal power-series ring
Lemma
A nondegenerate regular-trace pairing forces semisimplicity
Lemma
Matrix units lift through a finite free split formal deformation
Theorem
Finite Coxeter Hecke algebras are semisimple over the generic characteristic-zero field
Theorem
Split formal deformations preserve simple-block labels; the trace determinant bounds exceptional parameters
Sources
- Meinolf Geck, Modular Representations of Hecke Algebras, EPFL course notes
- Maarten Solleveld, Affine Hecke Algebras and Their Representations
- Benjamin Sambale, An Invitation to Formal Power Series, author-hosted full-text notes
- Herbert S. Wilf, generatingfunctionology, author-hosted second-edition PDF
- Pavel Etingof, Structure of Finite Dimensional Algebras, MIT OpenCourseWare 18.712, Chapter 7
- Arun Ram, Lectures in Representation Theory, Lecture 5