Alphabeta MathCyclotomic Hecke and Quiver Hecke Algebras
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
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Braid Groups
Categorification and Geometric Representation Theory
- Cyclotomic KLR Algebras and Highest-Weight Categorification
- Divided Powers, Integral Forms, and Quantum Highest Weights
- Kashiwara Operators and Crystal Lattices
- Kazhdan–Lusztig Bases, Polynomials, and Cells
- KLR Algebras, Induction, and Quantum-Serre Categorification
- Ordinary Hecke Specht Polytabloids and Signed Murphy Comparison
- Quantized Enveloping Algebras and Quantum Serre Relations
- Quantum Affine Type A and Level-One Fock Space
- Type-A Hecke Specht Modules and Cellular Bases
Category Theory
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Lie Theory
Linear Algebra
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Representation Theory of Groups
Set Theory
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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
- 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 Hook Length Formula and Rsk Correspondence
- Young Diagrams Tableaux and Permutation Modules
2 · Logical flowchart
3 · Items
Definition
Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules
Lemma
Laurent binomial valuation at one over any field
Lemma
Ultra-tight multisets admit integer column potentials
Definition
Affine type-A presentations and charged cyclotomic quotients
Lemma
DVR cellular Jantzen filtration and exact Gram valuation
Lemma
Universal nondegenerate affine Hecke ordered monomials form a basis
Lemma
Bounded ordered monomials span the cyclotomic Hecke quotients
Lemma
Degenerate affine Hecke ordered monomials form a basis
Lemma
Selfadjoint weight Gram valuations detect Jantzen classes
Lemma
Symmetric polynomial centers have primitive split content idempotents
Lemma
Polynomial spectra and central content sectors of finite cyclotomic Hecke algebras
Definition
Central content sectors of the cyclotomic Hecke algebra
Lemma
Charged core height matrices: hub-preserving descent and minimal switch connectivity
Lemma
Cyclic type-A KLR regular Hom norm gives the total dimension
Proposition
Cyclotomic Hecke algebras of level one are finite Hecke algebras
Definition
Affine and cyclotomic type-A Hecke algebras
Definition
The type-A cyclotomic specialization of the QG-KLR algebras
Lemma
Cyclotomic Hecke intertwiners and local power series
Lemma
The cyclotomic KLR block satisfies the Hecke relations
Lemma
Hecke intertwiners produce the local KLR generators
Theorem
Cyclotomic Hecke central sectors are isomorphic to type-A cyclotomic KLR algebras
Lemma
Basis and dimension of the cyclotomic Hecke algebra
Lemma
Universal cyclotomic Hecke ordered monomials form a basis
Lemma
The degenerate cyclotomic Hecke center is the symmetric polynomial image
Lemma
The universal cyclotomic Hecke canonical trace is perfect and symmetric
Theorem
The arbitrary-level cyclotomic Hecke Murphy basis is cellular
Lemma
Arbitrary-level Murphy Jucys–Murphy action is content triangular
Lemma
Cyclotomic semistandard row-type sums give permutation-module bases
Lemma
Cyclotomic semistandard sums give the actual Schur endomorphism basis
Theorem
Cyclotomic Schur standards and corner functor control block linkage
Lemma
Generic cyclotomic Schur deformation standards have nondegenerate forms
Lemma
Cyclotomic Schur weight Gram determinants are products of monic semistandard norms
Lemma
Cyclotomic Schur weight Gram valuations equal signed rim-hook Kostka sums
Theorem
Cyclotomic Schur Jantzen sum formula from rim-hook valuations
Lemma
Cyclotomic Jantzen equivalence is Schur and Hecke block linkage
Lemma
Multipartitions with fixed component cores are cyclotomic Jantzen-connected
Theorem
Cyclotomic multipartitions have equal residue content exactly when Jantzen-connected
Theorem
Central content sectors are zero or primitive cyclotomic Hecke blocks
Definition
Residue-sequence idempotents and cyclotomic Hecke blocks
Theorem
The Brundan–Kleshchev blockwise Hecke–KLR isomorphism
Corollary
Cyclotomic Hecke blocks inherit ℤ-gradings
Sources
- Mathas, The representation theory of the Ariki–Koike and cyclotomic q-Schur algebras, math/0204025
- Mathas, Hecke algebras and Schur algebras of the symmetric group, complete 79-page manuscript
- Lyle–Mathas, Blocks of cyclotomic Hecke algebras, math/0607451v4
- Mathas, Positive Jantzen sum formulas for cyclotomic Hecke algebras, arXiv:2106.15486v2
- Matthew Fayers, Core blocks of Ariki–Koike algebras, author manuscript, J. Algebraic Combin. 26 (2007), 47–81
- Brundan and Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, arXiv:0808.2032v4
- Jonathan Brundan, Centers of degenerate cyclotomic Hecke algebras and parabolic category O, arXiv:math/0607717v3
- Matthew Fayers, Weights of multipartitions and representations of Ariki–Koike algebras, author manuscript, Adv. Math. 206 (2006), 112–144
- Kang and Kashiwara, Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras, arXiv:1102.4677v4
- Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, arXiv:0808.2032v3, Invent. Math. 178 (2009) 451–484 (complete 32-page article)
- Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, arXiv:0909.4844v1, Bull. AMS 47 (2010) 419–481 (complete 66-page survey)
- Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete prepublication manuscript (AMS University Lecture Series 15, 1999), 79 pages
- Jun Hu and Andrew Mathas, Seminormal forms and cyclotomic quiver Hecke algebras of type A, arXiv:1304.0906v3 (published in Math. Ann.), complete 51-page article
- Jonathan Brundan and Alexander Kleshchev, Graded decomposition numbers for cyclotomic Hecke algebras, arXiv:0901.4450v3, Adv. Math. 222 (2009) 1883–1942 (complete 57-page article)
- Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups I, arXiv:0803.4121v3, Represent. Theory 13 (2009) 309–347 (complete 53-page article)
- Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, arXiv:0808.2032v4, Invent. Math. 178 (2009) 451–484 (complete 32-page article)
- Gunter Malle and Andrew Mathas, Symmetric cyclotomic Hecke algebras, J. Algebra 205 (1998), 275–293
- Mathas, Seminormal forms and Gram determinants for cellular algebras, math/0604108v4
- Jonathan Brundan, Centers of degenerate cyclotomic Hecke algebras and parabolic category O, math/0607717v3
- Brundan–Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras, arXiv:0808.2032v4
- Brundan and Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras, arXiv:0808.2032v3, Invent. Math. 178 (2009) 451–484 (complete 32-page article)
- Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras, arXiv:0808.2032v3, Invent. Math. 178 (2009) 451–484 (complete 32-page article)