Alphabeta MathType-A Hecke Specht Modules and Cellular Bases
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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
Category Theory
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
- 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 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
Cellular bases and cell modules
Definition
e-regular and e-restricted partitions
Definition
The quantum characteristic of an invertible Hecke parameter
Lemma
Decompositions of the identity into primitive central idempotents are unique
Lemma
The identity of a finite-dimensional algebra decomposes into primitive central idempotents
Lemma
The star anti-automorphism of the type-A Hecke algebra
Definition
Blocks of a finite-dimensional algebra via primitive central idempotents
Definition
Hecke Jucys-Murphy elements and q-contents
Definition
Murphy elements and Murphy tableau elements
Lemma
The cell form is associative and its radical is a submodule
Theorem
The Murphy tableau elements form a cellular basis
Theorem
Nonzero cell heads classify the simple modules of a cellular algebra
Corollary
Cellular decomposition matrices are unitriangular for the cell order
Definition
The type-A Hecke Specht module and its cell form
Definition
The Jantzen filtration of a lifted cell module
Theorem
The Hecke Jucys-Murphy action is dominance triangular
Lemma
Jantzen layer dimensions sum to the Gram determinant valuation
Theorem
Separated Hecke residues give seminormal idempotents and bases
Proposition
Hecke Specht theory specialises to group Specht theory at q = 1
Lemma
Nonzero-form cellular heads, cell purity and shared-factor linkage
Lemma
Type-A Hecke permutation modules: semistandard Murphy bases and Specht filtrations
Definition
The type-A q-Schur algebra and its Weyl modules
Definition
The Schur idempotent and the Schur functor for type-A q-Schur algebras
Lemma
Semistandard weight dimensions detect virtual Weyl classes
Definition
Virtual Weyl and Specht symbols attached to beta sequences
Theorem
The simple type-A Hecke modules are indexed by e-regular partitions
Corollary
The type-A Hecke decomposition matrix is dominance unitriangular
Theorem
The Schur functor sends Weyl and simple modules to Specht and Hecke simple modules
Theorem
The q-Schur Weyl weight-space Gram determinant is a hook product
Theorem
The q-Schaper virtual radical-layer formula
Remark
General type-A decomposition numbers remain unsolved
Lemma
Central Hecke Jucys–Murphy scalars separate q-Schur residue contents
Definition
The type-A quantum algebra and its tensor-action formulas
Lemma
The type-A quantum presentation descends to its coproduct tensor action
Lemma
Integral divided-power tensor actions generate the type-A q-Schur algebra
Lemma
Polynomial quantum highest vectors satisfy every divided-power highest relation
Lemma
Integral quantum highest modules are the normalized q-Schur Weyl cells
Lemma
Polynomial quantum Schur first levels: Weyl filtration and the actual Hecke corner
Sources
- Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete prepublication manuscript (1998; expanded published edition: AMS University Lecture Series 15, 1999); complete 79-page manuscript, SHA-256 8cfeb611a95a3ff25cb95ad22bfd3a68475818301c34b04d131a8480a1b997e0
- Andrew Mathas and Marcos Soriano, Seminormal forms and Gram determinants for cellular algebras, arXiv:math/0604108; J. reine angew. Math. 619 (2008) 141-173; complete 29-page author-hosted preprint, SHA-256 a2cd21ced7f35dee695388304c26804ef8ac9c54b661a5d624501c80864698a1
- Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, arXiv:0909.4844; Bull. Amer. Math. Soc. 47 (2010) 419-481; complete 66-page arXiv survey, SHA-256 8685199608967fa7c479e4dd883990b716a6f708e71cf75f6ce4aba41c6f2cab
- Jun Hu and Andrew Mathas, Seminormal forms and cyclotomic quiver Hecke algebras of type A, arXiv:1304.0906; complete 51-page preprint (v3, 2014), SHA-256 54b112868aef305ccbb8ba1608005ef84ee1457104ec41d3e69af6c22c2cf520
- Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, arXiv:0909.4844v1, complete 66-page survey
- Andrew Mathas, Cyclotomic quiver Hecke algebras of type A, arXiv:1310.2142; in: Modular Representation Theory of Finite and p-adic Groups, Lecture Notes Series 30, World Scientific (2015) 165-266; complete 54-page preprint
- Jun Hu and Andrew Mathas, Seminormal forms and cyclotomic quiver Hecke algebras of type A, arXiv:1304.0906, complete 51-page preprint
- Andrew Mathas, Seminormal forms and Gram determinants for cellular algebras, arXiv:math/0604108; J. reine angew. Math. 619 (2008) 141-173; complete 29-page author-hosted preprint, SHA-256 a2cd21ced7f35dee695388304c26804ef8ac9c54b661a5d624501c80864698a1
- Gordon James and Adalbert Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications 16, Cambridge University Press (1984 edition; originally Addison-Wesley, 1981)
- 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
- Gordon James and Andrew Mathas, A q-analogue of the Jantzen-Schaper theorem, Proc. London Math. Soc. (3) 74 (1997) 241-274
- Richard Dipper, Gordon James and Andrew Mathas, The (Q,q)-Schur algebra, Proc. London Math. Soc. (3) 77 (1998) 327-361; arXiv:q-alg/9701024
- Richard Dipper and Gordon James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986) 20-52
- Matthew Fayers, Irreducible Specht modules for Iwahori-Hecke algebras of type B, Representation Theory 16 (2012), 108-126
- Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, prepublication manuscript; Chapter 3, printed pp. 26–29, and Chapter 4, printed pp. 40–42, for the Murphy action and semistandard cell maps. The local proof uses an injective Weyl-cell map, not singular-parameter surjectivity onto all Hom spaces.
- Pavel Etingof and Mykola Semenyakin, A brief introduction to quantum groups
- Jonathan Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type A
- Jonathan Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type A, §§3.17–3.22 and 5.5–5.6