Alphabeta MathCyclotomic Murphy Bases and Cellular Module Constructions
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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 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
- 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
- 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
Categorification and Geometric Representation Theory
- Cyclotomic Hecke Quotients and Ariki–Koike PBW
- Generic Coxeter Hecke Algebras and the Standard Basis
- Kazhdan–Lusztig Bases, Polynomials, and Cells
- Ordinary Hecke Specht Polytabloids and Signed Murphy Comparison
- Tensor Coherence and Algebraic Descent
- Type A Affine Hecke Algebras and Bernstein PBW
- Type-A Hecke Specht Modules and Cellular Bases
Commutative Algebra
Coxeter Groups
Group Theory
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
- 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
Set Theory
2 · Logical flowchart
3 · Items
Definition
Cell data, multipartition dominance, and higher-cell ideals
Lemma
The cyclotomic anti-involution fixes the affine and Murphy generators
Definition
Cyclotomic Murphy elements
Lemma
Integral type-A Garnir straightening and the Murphy cell law
Lemma
Ordinary Murphy straightening preserves both tableau prefixes
Lemma
Componentwise initial-row straightening
Lemma
Cyclotomic component shift and multipartition dominance ideals
Theorem
The cyclotomic Murphy basis and cellularity
Theorem
Cell modules, their forms, and radicals
Theorem
Nonzero cellular heads classify the simple modules over a field
Sources
- Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete author-uploaded preliminary lecture manuscript (1998)
- Richard Dipper, Gordon James, and Andrew Mathas, Cyclotomic q-Schur algebras, Mathematische Zeitschrift 229 (1998), 385–416
- Andrew Mathas, Hecke algebras with cyclotomic parameters, in Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group, complete preliminary lecture manuscript
- Jun Hu and Andrew Mathas, Seminormal forms and cyclotomic quiver Hecke algebras of type A, arXiv:1304.0906v3 (2014)
- Richard Dipper, Gordon James, and Andrew Mathas, Cyclotomic q-Schur algebras
- Jun Hu and Andrew Mathas, Seminormal forms and cyclotomic quiver Hecke algebras of type A