Alphabeta MathOrdinary Hecke Specht Polytabloids and Signed Murphy Comparison
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1 · Prerequisites
Abstract Algebra
- Algebraic Extensions, Extension Degree, and Finite Fields
- 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, 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
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
Category Theory
Commutative Algebra
Coxeter Groups
Group Theory
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
2 · Logical flowchart
3 · Items
Definition
Hecke row permutation modules and ordinary Dipper-James polytabloids
Lemma
The row permutation module: distinguished basis and Hecke action
Lemma
The permutation form is H-contravariant and restricts to the DJ form
Lemma
The Hecke sign automorphism, pullback and contragredient dual
Lemma
Row symmetrizers and signed column symmetrizers vanish outside dominance
Lemma
The Murphy cellular basis, the cell law and the Murphy form
Lemma
Integral polytabloid straightening for the comparison
Lemma
Theta transports the Murphy cellular structure to the signed cell modules
Theorem
Murphy cells are the same-shape contragredient Specht duals
Theorem
Ordinary Specht modules are the signed conjugate Murphy cells
Corollary
Polytabloid specialization and the exact signed multiplicity dictionary
Sources
- Matthew Fayers, An algorithm for semistandardising homomorphisms, author version accepted by J. Algebra, 16 PDF pages
- Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, author-hosted prepublication manuscript, 79 PDF pages
- Samuel David Heard, Jonathan R. Kujawa, Positivity and web bases for Specht modules of Hecke algebras, arXiv:2207.00654v2, 26 PDF pages
- R. Dipper, G. D. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986) 20-52, DOI 10.1112/plms/s3-52.1.20
- Steen Ryom-Hansen, On the denominators of Young's seminormal basis, arXiv:0904.4243v5, 28 PDF pages
- Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, arXiv:0909.4844v1 (survey), 66 PDF pages
- Alexander S. Kleshchev, Andrew Mathas, Arun Ram, Universal graded Specht modules for cyclotomic Hecke algebras, author-hosted Sydney preprint, 44 PDF pages
- G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer 1978; author-hosted scan