Alphabeta MathThe Weyl Kac Character Formula
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1 · Prerequisites
Abstract Algebra
- 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
Commutative Algebra
Lie Theory
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- 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
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- 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
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- 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
2 · Logical flowchart
3 · Items
Definition
Kac Moody formal character completion
Definition
Kac Moody Weyl vector
Definition
Kac Moody denominator product with root multiplicities
Lemma
The Kac Moody denominator is Weyl skew
Lemma
Casimir constrained Verma character expansion
Lemma
The denominator quotient has only imaginary cone support
Lemma
Casimir norm excludes nonzero denominator corrections
Theorem
Kac Moody denominator identity
Lemma
The shifted integrable character numerator is Weyl skew
Lemma
Only the highest dot orbit can occur in the integrable numerator
Theorem
Weyl Kac character formula
Definition
Generalized Kostant partition function
Corollary
Kac Moody Kostant multiplicity formula
Corollary
Weyl Kac specializes to the finite Weyl character formula
Proposition
Affine denominator separates real and imaginary root factors
Remark
Weyl Kac products are formal not analytic identities here
Sources
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Section 9.2
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Section 10.7
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Sections 2.3 and 10.1
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Chapter 11
- Kleshchev, Sections 9.2 and 10.2
- Perrin, Sections 10.7 and 11.2
- Kleshchev, Lemma 10.1.1 and Section 10.2
- Perrin, Section 11.2
- Kleshchev, Proposition 9.2.5
- Perrin, Lemma 11.2.4
- Kleshchev, Sections 5.3 and 10.1-10.2
- Kleshchev, Proposition 9.2.5 and Theorem 10.2.1
- Perrin, Lemmas 11.2.4–11.2.5 and Theorem 11.2.1
- Kleshchev, Theorem 10.2.1 at Lambda=0
- Perrin, Theorem 11.2.1 at lambda=0
- Kleshchev, Section 10.1
- Kleshchev, Lemma 10.1.2 and Theorem 10.2.1
- Perrin, Lemma 11.2.5 and Theorem 11.2.1
- Kleshchev, Theorem 10.2.1
- Perrin, Theorem 11.2.1
- Kleshchev, Corollary 10.2.2
- Kleshchev, Theorem 10.2.1 and finite-type discussion
- Kleshchev, Sections 6.1-6.3 and 10.3
- Perrin, Chapter 12