Alphabeta MathHecke Parabolic Induction and Symmetrizing Traces
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 2 results · all verified · 1 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 1 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
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
Parabolic Hecke subalgebras, induction, and restriction
Definition
The canonical Hecke trace and symmetrizing form
Theorem
Parabolic freeness and bases of induced Hecke modules
Theorem
The canonical Hecke trace is symmetric and perfect
Sources
- Christoph Schönnenbeck, Representations of Iwahori-Hecke Algebras Induced from Parabolic Subalgebras (RWTH Aachen Master's thesis, 2016)
- George Lusztig, Hecke Algebras with Unequal Parameters (revised book text, arXiv:math/0208154v2)
- Meinolf Geck, Modular Representations of Hecke Algebras (EPFL lecture notes, arXiv:math/0511548v2)
- Andrew Mathas, Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group (self-contained notes, 1998; complete 79-page text)