Alphabeta MathKazhdan–Lusztig Bases, Polynomials, and Cells
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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
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
Definition
The normalized type-A Hecke algebra and its bar involution
Lemma
The Hecke bar involution is well defined
Lemma
Basic properties of the Bruhat order on Sₙ
Definition
Bruhat intervals and the R-coefficients
Lemma
Reversal anti-involution commutes with the Hecke bar
Theorem
The R-coefficient recursion, support, degree bounds and inversion
Lemma
The type-A Kazhdan–Lusztig basis has elementary triangular bar properties
Definition
Kazhdan–Lusztig polynomials in the classical q-normalization
Theorem
Multiplication by a generator in the Kazhdan–Lusztig basis
Definition
Inverse Kazhdan–Lusztig polynomials
Lemma
Verma's sign identity over Bruhat intervals
Theorem
The Kazhdan–Lusztig polynomial descent recursion
Theorem
The Kazhdan–Lusztig inversion formula
Definition
L-, R- and two-sided Kazhdan–Lusztig preorders and cells
Definition
Knuth and dual Knuth equivalence for permutations
Theorem
Knuth classes are the fibers of the insertion tableau
Definition
Star operations on strings of adjacent simple reflections
Lemma
Star operations are Knuth moves and preserve the relevant cells
Lemma
μ-edges and left equivalence are transported by star operations
Proposition
Equal insertion or recording tableaux imply right or left equivalence
Lemma
Left equivalence forces equality of recording tableaux in type A
Theorem
Kazhdan–Lusztig cells of type A are classified by RSK tableaux
Sources
- Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, arXiv:1212.0791 (45 pp.) — §3.2 (printed pp. 15–16): the Hecke algebra in the normalization $H_xH_s=H_{xs}$ or $(v^{-1}-v)H_x+H_{xs}$, the bar involution $\overline{H_x}=H_{x^{-1}}^{-1}$, the Kazhdan–Lusztig basis $\{\underline H_x\}$ characterized by bar-invariance and $\underline H_x\in H_x+\sum_{y<x}v\mathbb Z[v]H_y$, the example $\underline H_s=H_s+vH_{\mathrm{id}}$, and Remark 3.2: $v=q^{-1/2}$, $H_x=v^{\ell(x)}T_x$, $\underline H_x=C'_x$, $h_{y,x}=v^{\ell(x)-\ell(y)}P_{y,x}(v^{-2})$
- G. Lusztig, Hecke Algebras with Unequal Parameters (revised book version, arXiv:math/0208154v2) — the split case $L\equiv1$ read as the source for the bar operator, the R-coefficients, the new basis, its multiplication properties and cells; translated to the normalization of this page by $v_L=v^{-1}$ (so $v_L^{L(w)-L(y)}=v^{-(\ell(w)-\ell(y))}$)
- Arun Ram, Notes on Schubert Polynomials, Chapter 1: Permutations
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 (18 pp.) — the direct proof of the Kazhdan–Lusztig cell classification in type A via Knuth relations and transported Kazhdan–Lusztig graph edges
- G. Lusztig, Hecke Algebras with Unequal Parameters (revised book version, arXiv:math/0208154v2) — §4.3–4.9 (printed pp. 25–27): R-coefficients, descent recursion, support and degree bounds, Verma's sign sum, bar and inversion identities; full argument read. Section 4.3 writes bar(T_w)=sum_y bar(r^L_{y,w})T_y, so this page's direct expansion coefficients are bar(r^L_{y,w}); the parameter translation is v_L=v^{-1}.
- Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, arXiv:1212.0791 (45 pp.) — §3.2 (printed pp. 15–16): the normalized Hecke multiplication and bar conventions.
- G. Lusztig, Hecke Algebras with Unequal Parameters (revised book version)
- Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, arXiv:1212.0791 — §3.2, printed pp. 15–16: the Hecke normalization and Remark 3.2, with q=v^-2, H_x=v^ell(x)T_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 2.3 and Lemma 2.5(1): the classical q-polynomial normalization, degree bound, and constant term.
- G. Lusztig, Hecke Algebras with Unequal Parameters, revised version arXiv:math/0208154v2 — Theorem 5.2 and Proposition 5.4 in the split case L=1: the new-basis coefficients and degree/parity bounds.
- G. Lusztig, Hecke Algebras with Unequal Parameters, revised version arXiv:math/0208154v2 — §§6.3–6.7: the equal-parameter generator formulas; Corollary 6.5 identifies μ^s as the coefficient of v_L^{-1}, which becomes this page's coefficient of v under v_L=v^{-1}; Theorem 6.6 gives the left formula.
- Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, arXiv:1212.0791 — §3.2, printed pp. 15–16: the normalized Hecke algebra and Kazhdan–Lusztig basis.
- G. Lusztig, Hecke Algebras with Unequal Parameters, revised version arXiv:math/0208154v2 — §10.1–10.2: chain formula and inverse matrices; §10.7: the dual-basis interpretation.
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 (18 pp.) — §2.2: the classical Kazhdan–Lusztig polynomial and μ-coefficient conventions used with the sign normalization.
- G. Lusztig, Hecke Algebras with Unequal Parameters (revised book version, arXiv:math/0208154v2) — Proposition 4.8 (printed p. 27): Verma's alternating-sign sum over Bruhat intervals, derived from the R-matrix identity and the lowest-degree term of the R-coefficients.
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 — §2.2, Definition 2.4: the classical Kazhdan–Lusztig polynomial descent recursion and its μ-correction term.
- G. Lusztig, Hecke Algebras with Unequal Parameters, revised version arXiv:math/0208154v2 — §§6.1–6.7: equal-parameter Kazhdan–Lusztig basis multiplication and coefficient symmetry, translated to v_L=v^{-1}.
- G. Lusztig, Hecke Algebras with Unequal Parameters, revised version arXiv:math/0208154v2 — §10.1–10.2: the inverse chain formula, inverse-matrix identities, and signed convention; §10.7: the dual-basis interpretation.
- G. Lusztig, Hecke Algebras with Unequal Parameters (revised book version, arXiv:math/0208154v2) — §8.1 defines the left, right, and two-sided relations and cells; §§8.4–8.6 prove descent-set monotonicity. The equal-parameter case is translated to this normalization.
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 (18 pp.) — type-A Robinson–Schensted and Kazhdan–Lusztig cell conventions.
- Lars Thorge Jensen, p-Kazhdan–Lusztig Theory (Bonn dissertation 2017/18) — Kazhdan–Lusztig cell and star-operation conventions.
- Donald E. Knuth, Permutations, Matrices, and Generalized Young Tableaux, Pacific J. Math. 34 (1970), 709–727
- Lars Thorge Jensen, p-Kazhdan–Lusztig Theory (Bonn dissertation 2017/18), — the star operations, their action on structure coefficients, and the transfer of the type-A classification; his normalization is translated to the one of this page
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 — Definition 3.2, Theorem 3.3, and the complete proof of Lemma 3.4: Knuth moves as star operations and the corresponding cell relation.
- G. Lusztig, Hecke Algebras with Unequal Parameters (revised version arXiv:math/0208154v2) — Corollary 6.7 and the rank-two equal-parameter calculation in Proposition 7.3.
- Donald E. Knuth, Permutations, Matrices, and Generalized Young Tableaux, Pacific J. Math. 34 (1970), 709–727 — Theorem 6 and the two local transformations.
- Bill Casselman, Notes on Kazhdan–Lusztig polynomials — the right-descent recursion, mixed-descent vanishing, and the full star-operation leading-coefficient argument.
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 — star operations, nonvanishing transport, and cell transport.
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 — Proposition 3.8 and its complete proof via Knuth moves and star operations.
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 — the hard direction of Theorem A and the descent/block comparison.
- Meinolf Geck, Kazhdan–Lusztig cells and the Murphy basis, arXiv:math/0504217v2 — Corollary 5.6(c), printed p.29: only the implication from two-sided equivalence to equality of RSK shapes is cited under the exact owner authorization.
- Susumu Ariki, Robinson–Schensted correspondence and left cells, arXiv:math/9910117 — type-A left-cell classification via Knuth paths and transported star operations.