Math 2.0

Nonassociative rings and algebras Infinite-dimensional Lie algebras Theorem

Kac–Weyl character formula

Builds on Kac–Moody algebraGeneralized Cartan matrixHighest weightWeyl groupRepresentation of a Lie algebraWeyl character formulaAffine Lie algebraJacobi triple product

Statement

Let AA be a symmetrizable generalized Cartan matrix and let 𝔤=𝔤(A)\mathfrak{g} = \mathfrak{g}(A) be its Kac–Moody algebra. Let 𝔥\mathfrak{h} be its Cartan subalgebra, with simple roots α1,…,αn∈𝔥∗\alpha_1, \ldots, \alpha_n \in \mathfrak{h}^* and simple coroots α1∨,…,αn∨∈𝔥\alpha_1^\vee, \ldots, \alpha_n^\vee \in \mathfrak{h}. Let Δ+\Delta_+ be the set of positive roots, and for α∈Δ+\alpha \in \Delta_+ let mult⁡α\operatorname{mult} \alpha be the dimension of the root space 𝔤α\mathfrak{g}_\alpha. Let WW be the Weyl group, generated by the reflections si(λ)=λ−λ(αi∨) αis_i(\lambda) = \lambda - \lambda(\alpha_i^\vee)\, \alpha_i. For w∈Ww \in W let ε(w)=(−1)ℓ(w)\varepsilon(w) = (-1)^{\ell(w)}, where ℓ(w)\ell(w) is the length of ww in these generators. Fix ρ∈𝔥∗\rho \in \mathfrak{h}^* with ρ(αi∨)=1\rho(\alpha_i^\vee) = 1 for all ii. Let λ∈𝔥∗\lambda \in \mathfrak{h}^* be dominant integral, that is, λ(αi∨)∈ℤ≥0\lambda(\alpha_i^\vee) \in \Z_{\ge 0} for all ii. Let L(λ)L(\lambda) be the irreducible highest weight module of 𝔤\mathfrak{g} with highest weight λ\lambda. Write ch⁡L(λ)=∑μ∈𝔥∗dim⁡L(λ)μ eμ\operatorname{ch} L(\lambda) = \sum_{\mu \in \mathfrak{h}^*} \dim L(\lambda)_\mu\, e^\mu for its formal character. Then ch⁡L(λ)=∑w∈Wε(w) ew(λ+ρ)−ρ∏α∈Δ+(1−e−α)mult⁡α.\operatorname{ch} L(\lambda) = \frac{\sum_{w \in W} \varepsilon(w)\, e^{w(\lambda + \rho) - \rho}}{\prod_{\alpha \in \Delta_+} \left(1 - e^{-\alpha}\right)^{\operatorname{mult} \alpha}}.

Remarks

The proof follows the algebraic proof of the finite-dimensional Weyl character formula. The product ch⁡L(λ)⋅∏α∈Δ+(1−e−α)mult⁡α\operatorname{ch} L(\lambda) \cdot \prod_{\alpha \in \Delta_+} (1 - e^{-\alpha})^{\operatorname{mult} \alpha} is anti-invariant under WW, after the shift by ρ\rho. A generalized Casimir operator, built from an invariant bilinear form that exists because AA is symmetrizable, shows that only the weights w(λ+ρ)−ρw(\lambda + \rho) - \rho can occur in it. Anti-invariance then fixes the coefficient of each such weight as ε(w)\varepsilon(w).

For λ=0\lambda = 0 the module L(0)L(0) is trivial, and the formula becomes the denominator identity ∏α∈Δ+(1−e−α)mult⁡α=∑w∈Wε(w) ewρ−ρ.\prod_{\alpha \in \Delta_+} \left(1 - e^{-\alpha}\right)^{\operatorname{mult} \alpha} = \sum_{w \in W} \varepsilon(w)\, e^{w\rho - \rho}. For affine Lie algebras these denominator identities are the Macdonald identities, and for the affine algebra of 𝔰𝔩2\mathfrak{sl}_2 the identity is equivalent to the Jacobi triple product.

The hypothesis that λ\lambda is dominant integral is necessary. For example, let 𝔤\mathfrak{g} be finite-dimensional and λ=−ρ\lambda = -\rho, which is not dominant integral. Then the numerator is ∑w∈Wε(w) e−ρ=0\sum_{w \in W} \varepsilon(w)\, e^{-\rho} = 0, because WW has as many elements of sign 11 as of sign −1-1, while L(−ρ)≠0L(-\rho) \neq 0 has a nonzero character.

Going deeper

For a student

When 𝔤\mathfrak{g} is finite-dimensional, all multiplicities are 11, the Weyl group is finite, and the formula is Weyl's character formula. For affine Lie algebras, specializing the formal exponentials to powers of a variable qq turns both sides into qq-series, and the formula produces identities between infinite products and theta-type sums. This is the source of the link between affine Lie algebras and modular forms. The theorem was later proved for all generalized Cartan matrices, without the symmetrizability hypothesis, by Kumar and by Mathieu.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

Show the path as a graph
Affine Lie algebra, concept, no evidence levelAffine Lie algebraconceptCartan subalgebra, concept, no evidence levelCartan subalgebraconceptComplex numbers, object, no evidence levelComplex numbersobjectCoroot, concept, no evidence levelCorootconceptGeneralized Cartan matrix, concept, no evidence levelGeneralized Cartan ma…conceptHighest weight, concept, no evidence levelHighest weightconceptIrreducible Lie module, concept, no evidence levelIrreducible Lie moduleconceptKac–Moody algebra, concept, no evidence levelKac–Moody algebraconceptRepresentation of a Lie algebra, concept, no evidence levelRepresentation of a L…conceptSemisimple Lie algebra, concept, no evidence levelSemisimple Lie algebraconceptSequence, definition, no evidence levelSequencedefinitionSimple roots, concept, no evidence levelSimple rootsconceptWeight of a representation, concept, no evidence levelWeight of a represent…conceptWeyl group, concept, no evidence levelWeyl groupconceptq-Pochhammer symbol, concept, no evidence levelq-Pochhammer symbolconceptTheorem of the highest weight, statement, no evidence levelTheorem of the highes…statementInfinite product, definition, no evidence levelInfinite productdefinitionWeyl character formula, statement, no evidence levelWeyl character formulastatementJacobi triple product, statement, no evidence levelJacobi triple productstatementKac–Weyl character formula, statement, no evidence levelKac–Weyl character fo…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Affine Lie algebra concept
  2. Cartan subalgebra concept
  3. Complex numbers object
  4. Coroot concept
  5. Generalized Cartan matrix concept
  6. Highest weight concept
  7. Irreducible Lie module concept
  8. Kac–Moody algebra concept
  9. Representation of a Lie algebra concept
  10. Semisimple Lie algebra concept
  11. Sequence definition
  12. Simple roots concept
  13. Weight of a representation concept
  14. Weyl group concept
  15. q-Pochhammer symbol concept
  16. Theorem of the highest weight statement
  17. Infinite product definition
  18. Weyl character formula statement
  19. Jacobi triple product statement
  20. Kac–Weyl character formula statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Weyl_character_formula
About this page
TagRJKHTWHME1
URIm2://RJKHTWHME1
Kindstatement, theorem
WritingW0 stub
MSC17B67, 17B10
StatusThis copy of the site has no build data. Run m2 status RJKHTWHME1.
AboutKac–Moody algebra
UsesKac–Moody algebra, Generalized Cartan matrix, Highest weight, Weyl group, Representation of a Lie algebra, Weyl character formula, Affine Lie algebra, Jacobi triple product
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by