Alphabeta MathLevel-One Fock Space and Residue Crystals
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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 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 over a Principal Ideal Domain and the Canonical Forms
- 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
- Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Braid Groups
Categorification and Geometric Representation Theory
- Balanced Triples and Global Crystal Bases
- Divided Powers, Integral Forms, and Quantum Highest Weights
- Kashiwara Operators and Crystal Lattices
- Ordinary Hecke Specht Polytabloids and Signed Murphy Comparison
- Quantized Enveloping Algebras and Quantum Serre Relations
- Quantum Affine Type A and Level-One Fock Space
- Tensor Coherence and Algebraic Descent
- Tensor-Product Crystals and Highest-Weight Crystal Uniqueness
- Type-A Hecke Specht Modules and Cellular Bases
Commutative Algebra
Coxeter Groups
Group Theory
Homological Algebra
Lie Theory
- Finite Weyl Invariants, Bruhat Order, and Kostant Harmonics
- Harish Chandra Isomorphism Casimir and Central Characters
- Integrable Highest Weight Kac Moody Modules
- Kac Moody Algebras from Generalized Cartan Matrices
- Lie Algebra Representations, Enveloping Algebras, and PBW
- Tensor Product Multiplicities and Littlewood Richardson
- The Weyl Kac Character Formula
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
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- 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
- Constant Rank, Submersions, Immersions and Regular Level Sets
- 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
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Representation Theory of Groups
Set Theory
Symmetric Groups and Symmetric Functions
- Cores, Quotients, and Blocks of Symmetric Groups
- Integral Specht Modules and Modular Simple Modules
- Jucys–Murphy Elements and Seminormal Forms
- Modular Branching and the Good-Node Crystal
- Rim Hooks and the Murnaghan–Nakayama Rule
- Specht Modules and the Irreducibles of the Symmetric Group
- Symmetric Functions, the Hall Inner Product, and Schur Bases
- The Branching Rule and the Young Graph
- The Hook Length Formula and Rsk Correspondence
- Young Diagrams Tableaux and Permutation Modules
Topology
2 · Logical flowchart
3 · Items
Definition
The symmetric-group level-one residue and partition-label convention
Lemma
Residue-node reading orders and the conjugation dictionary
Proposition
The QG Fock-crystal arrows are the modular good-node operators
Theorem
The restricted-partition component is the basic crystal in the symmetric-group convention
Definition
Level-one global-basis transition polynomials in the symmetric-group convention
Lemma
General Hecke cellular simple labels match the basic crystal
Sources
- Alexander Kleshchev, Representation Theory of Symmetric Groups and Related Hecke Algebras, Bull. AMS 47 (2010) 419-481, arXiv:0909.4844 (complete 66-page survey)
- Matthew Fayers, Partition models for the crystal of the basic U_q(sl_n)-module, arXiv:0906.4129 (complete 31-page paper)
- Bernard Leclerc, Fock space representations (complete author-hosted 38-page lecture notes)
- M. Kashiwara, T. Miwa, J.-U. H. Petersen, C. M. Yung, Perfect crystals and q-deformed Fock spaces, arXiv:q-alg/9603025 (complete 74-page paper)
- Bernard Leclerc and Jean-Yves Thibon, Canonical bases of q-deformed Fock spaces, arXiv:q-alg/9602025 (complete 8-page paper)
- Jonathan Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type A, §§2.1–2.11 and 5.3–5.7
- Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, §11.2