Alphabeta MathProjective Extensions and the Little Group Method
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Characters and the Orthogonality Relations
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Averaging and Character-Theory Prerequisites
- 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
- Induced Representations, Frobenius Reciprocity and Applications
- Maschke's Theorem, Complete Reducibility and the Structure of k[G]
- 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
- Semidirect Products, Automorphism Groups and Split Extensions
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Tensor Products of Modules
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite 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
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- 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
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- 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
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Representation Theory of Groups
2 · Logical flowchart
3 · Items
Definition
Projective representations and normalized factor sets
Lemma
The factor set satisfies the two-cocycle equation
Lemma
Rephasing changes factor sets by coboundaries
Definition
Twisted group algebra of a factor set
Lemma
Projective representations and twisted algebra modules
Lemma
An invariant irreducible normal representation yields projective inertia operators
Definition
The Clifford obstruction class of an invariant irreducible representation
Theorem
An invariant irreducible representation extends to its inertia group exactly when the Clifford obstruction vanishes
Lemma
The twisted product of a normalized cocycle is a central extension
Lemma
The cocycle central extension linearizes a projective representation
Theorem
The projective Clifford correspondence for an invariant irreducible representation
Theorem
The little group method for a semidirect product with abelian kernel
Sources
- Britta Späth, Reduction theorems for some global-local conjectures — Definition 1.4 and Remark 1.5, printed pp. 2–3
- Tammo tom Dieck, Representation Theory — §4.2, printed pp. 54–57
- Britta Späth, Reduction theorems for some global-local conjectures — Definition 1.4 and the following cocycle identity, printed p. 3
- Clara Loh, Group Cohomology, SS 2019 — normalized two-cocycles
- Britta Späth, Reduction theorems for some global-local conjectures — Remark 1.5(a), printed p. 3
- Clara Loh, Group Cohomology, SS 2019 — coboundaries and H^2
- Britta Späth, Reduction theorems for some global-local conjectures — Remark 1.5(b), printed p. 3
- Peter Webb, A Course in Finite Group Representation Theory — semisimplicity of finite group algebras
- Britta Späth, Reduction theorems for some global-local conjectures — Definition 1.4, Lemma 1.8(a)–(d), printed pp. 2–4
- Britta Späth, Reduction theorems for some global-local conjectures — §1.B (character triples, associated projective representations, Lemma 1.8(d)), printed pp. 3–4
- Britta Späth, Reduction theorems for some global-local conjectures — Lemma 1.8(d) and the discussion after Remark 1.9, printed pp. 3–4
- Tammo tom Dieck, Representation Theory — Remarks (4.2.5)–(4.2.6), printed p. 57
- Britta Späth, Reduction theorems for some global-local conjectures — Proposition 1.11, printed p. 4
- Britta Späth, Reduction theorems for some global-local conjectures — Theorem 1.12(b), printed pp. 4–5
- Britta Späth, Reduction theorems for some global-local conjectures — Theorem 1.12 and Corollary 1.13, printed pp. 4–5
- Tammo tom Dieck, Representation Theory — §(4.2.4)–(4.2.7), printed pp. 56–57
- Tammo tom Dieck, Representation Theory — Propositions (4.2.4), (4.2.6) and Remark (4.2.7), printed pp. 56–57
- Britta Späth, Reduction theorems for some global-local conjectures — Theorem 1.3 (Gallagher) and Theorem 1.2 (Clifford), printed pp. 2–3