Alphabeta MathModular Traces and Brauer-Character Independence
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1 · Prerequisites
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inverse Limits and Noetherian Completion
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Modular Representations and Projective Covers
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
- The ZFC Axioms and the Basic Set Constructions
- Valuation Rings and Discrete Valuation Rings
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Lemma
A finite-dimensional algebra separates its split simple modules
Lemma
Trace functionals of split simple modules are independent
Lemma
Modular trace depends only on the p-regular part
Lemma
Prime-to-p roots lift uniquely in a complete DVR
Definition
Lifted modular trace on p-regular elements
Lemma
Reduction of lifted traces recovers modular traces
Theorem
Irreducible Brauer characters are independent on p-regular elements
Sources
- Yanqi Lake Lectures on Algebra I, Theorems 11.1.5 and 11.2.2, pp.128 and 130
- Pound/Martin, Modular Representation Theory, Lemma 6.6, p.18
- Stacks Project, Lemmas 10.153.2 and 10.153.9
- Webb, A Course in Finite Group Representation Theory, Section 10.1 pp.169–171 and Theorem 10.2.2 p.176
- Halle, Galois actions on Neron models of Jacobians, Section 5.3, p.875