Alphabeta MathKazhdan–Lusztig Bases, Polynomials, and Cells — Examples
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1 · Prerequisites
Abstract Algebra
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- 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
Categorification and Geometric Representation Theory
Coxeter Groups
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- 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
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
Representation Theory of Groups
Set Theory
Symmetric Groups and Symmetric Functions
2 · Logical flowchart
3 · Items
Example
The Kazhdan–Lusztig bases of S₂ and S₃
Example
The R- and Kazhdan–Lusztig recursions on a small singular interval
Example
RSK cells in S₃ and S₄
Sources
- Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, arXiv:1212.0791 — §3.2 (printed pp. 15–16): the normalized Hecke algebra, bar involution, triangular Kazhdan–Lusztig basis, rank-one element, and the dictionary $\underline H_x=C'_x$ and $h_{y,x}=v^{\ell(x)-\ell(y)}P_{y,x}(v^{-2})$.
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 — §2.2, Definition 2.4: the descent recursion for Kazhdan–Lusztig polynomials and its μ-correction term.
- G. Lusztig, Hecke Algebras with Unequal Parameters, revised version arXiv:math/0208154v2 — §§4.3–4.9: R-coefficients, recurrence and bounds; §§10.1–10.2: the inverse chain formula and inverse matrices, in the equal-parameter specialization.
- Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, arXiv:1212.0791 — §3.2 (printed pp. 15–16): normalized Hecke algebra, bar and KL-basis conventions, and Remark 3.2 with $\underline H_x=C'_x$ and $h_{y,x}=v^{\ell(x)-\ell(y)}P_{y,x}(v^{-2})$.
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 (18 pp.)