Alphabeta MathQuantum Affine Type A and Level-One Fock Space — Examples
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- 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 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
- Tensor Products of Modules
- The Field of Fractions and Localisation
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
- Quantized Enveloping Algebras and Quantum Serre Relations
- Quantum Affine Type A and Level-One Fock Space
- Tensor-Product Crystals and Highest-Weight Crystal Uniqueness
Commutative Algebra
Coxeter Groups
Lie 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
- Exterior Powers, Orientation and Hodge Duality
- 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
Real Analysis
Set Theory
Symmetric Groups and Symmetric Functions
2 · Logical flowchart
3 · Items
Example
Node actions for partitions of size at most three
Example
The double-edge Serre cancellation in affine A₁ on partitions
Example
The good-node crystal component of the empty partition for e=3
Counterexample
The whole partition Fock space is not the basic representation
Example
Unitriangular global basis elements at small rank
Sources
- Bernard Leclerc, Fock space representations (complete author-hosted 38-page body)
- Bernard Leclerc and Jean-Yves Thibon, Canonical bases of q-deformed Fock spaces, arXiv:q-alg/9602025
- Matthew Fayers, Partition models for the crystal of the basic $U_q(\mathfrak{sl}_n)$-module, arXiv:0906.4129
- Masaki Kashiwara, Tetsuji Miwa and Eugene Stern, Decomposition of q-deformed Fock spaces, arXiv:q-alg/9508006
- Kashiwara-Miwa-Petersen-Yung, Perfect crystals and q-deformed Fock spaces
- Kailash C. Misra and Tetsuji Miwa, Crystal base for the basic representation of $U_q(\mathfrak{sl}(n))$, Comm. Math. Phys. 134 (1990), 79-88