Alphabeta MathFinite Weyl Invariants, Bruhat Order, and Kostant Harmonics
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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
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- 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 Field of Fractions and Localisation
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
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- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Commutative Algebra
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Depth and Cohen Macaulay Modules
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
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- Rees Modules Artin Rees and Hilbert Samuel Theory
Homological Algebra
Lie Theory
Linear Algebra
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- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
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- Rⁿ as a Normed Space; Vector-Valued Functions
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- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
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- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
2 · Logical flowchart
3 · Items
Definition
Finite Weyl root system, lattice and chamber conventions
Lemma
Finite Weyl positive roots and simple reflections
Lemma
Finite Weyl strong exchange and deletion
Lemma
Finite Weyl closed chambers and stabilizers
Definition
Bruhat order on a finite Weyl group
Definition
Weyl orbit sum in a group algebra
Lemma
Weyl orbit sums form a basis of finite Weyl invariants
Definition
Finite linear invariant and coinvariant polynomial algebras
Lemma
Finite linear group invariant polynomials separate orbits
Definition
Weyl discriminant and reflecting hyperplane arrangement
Lemma
Weyl anti invariants are divisible by the discriminant
Lemma
Finite reflection invariant generators are algebraically independent
Lemma
Reflection basic invariants form a regular sequence
Lemma
Weyl coinvariant hilbert series has order w dimension
Theorem
Chevalley shephard todd for finite weyl groups
Definition
Finite semisimple Lie algebras and the symmetric adjoint action
Lemma
Engel, the trace criterion, and Killing nondegeneracy
Lemma
Finite Lie triangularization and rank-one complete reducibility
Lemma
Finite semisimple Cartan, root and string structure
Lemma
Finite semisimple PBW and highest-weight construction
Definition
Kostant harmonic subspace of the symmetric algebra
Lemma
Highest-weight characters are unitriangular in Weyl orbit sums
Lemma
Kostant harmonics give an invariant polynomial complement
Lemma
Local Chevalley restriction for Kostant freeness
Theorem
Kostant harmonic decomposition of the symmetric algebra
Lemma
Filtered freeness lifts from associated graded algebras
Sources
- Pavel Etingof, Lie Groups and Lie Algebras, §§21–22; local sign-change proofs fill the chamber argument
- Pavel Etingof, Representations of Lie Groups, §§11–13; local proof and exact reading limits in the group report
- Pavel Etingof, Lie Groups and Lie Algebras, §§15–17; finite-dimensional local trace proof
- Pavel Etingof, Lie Groups and Lie Algebras, §§11,15; local generalized-weight splitting proof
- Pavel Etingof, Lie Groups and Lie Algebras, §19; local Cartan and root-string proof
- Pavel Etingof, Lie Groups and Lie Algebras, §§13,25; local finite PBW and integrable-quotient proof
- Pavel Etingof, Lie Groups and Lie Algebras, §§25–26; local orbit-sum triangular inversion
- Pavel Etingof, Lie Groups and Lie Algebras, §41.1; local universal sign-conjugation and positive Fischer proof
- Pavel Etingof, Representations of Lie Groups, Theorem10.1 pp54–55; local polynomial-density and finite character proof
- Pavel Etingof, Representations of Lie Groups, Theorem 13.1, first proof paragraph; explicit lifting and harmonic identification below
- Etingof, Representations of Lie Groups, Theorem 13.1, filtered lifting argument