Weyl Character Formula — Mathematics Knowledge System

Overview

The Weyl character formula is a closed-form expression for the character of a finite-dimensional irreducible representation V_λ of a complex semisimple Lie algebra (or compact Lie group). It expresses the character as a ratio of two alternating sums over the Weyl group, weighted by dominant weights. A specialisation yields the Weyl dimension formula. The proof uses the theory of Verma modules or the Weyl integration formula.

Intuition

The character χ_λ encodes all information about V_λ (its weight multiplicities). The formula expresses χ_λ as an 'alternating sum' over Weyl group elements, divided by the Weyl denominator. The numerator is the 'numerator character', and the denominator is a universal factor that appears in all characters.

Formal Definition

Definition

Let g be a complex semisimple Lie algebra with Cartan subalgebra h, root system Φ, positive roots Φ⁺, Weyl group W, and Weyl vector ρ = (1/2)Σ_{α>0} α. For a dominant integral weight λ ∈ Λ⁺, the character of the irreducible representation V_λ is the function on the regular elements of h given by:

\chi_\lambda = \frac{\sum_{w \in W} (-1)^{\ell(w)} e^{w(\lambda+\rho)}}{\sum_{w \in W} (-1)^{\ell(w)} e^{w(\rho)}} = \frac{A_{\lambda+\rho}}{A_\rho}

Weyl character formula

A_\mu = \sum_{w \in W} (-1)^{\ell(w)} e^{w(\mu)} \quad (\text{alternating sum})

Alternating sum over Weyl group

A_\rho = \prod_{\alpha \in \Phi^+} (e^{\alpha/2} - e^{-\alpha/2})

Weyl denominator formula

\dim V_\lambda = \prod_{\alpha \in \Phi^+} \frac{\langle \lambda + \rho,\, \alpha^\vee \rangle}{\langle \rho,\, \alpha^\vee \rangle}

Weyl dimension formula (specialisation)

Notation

Notation	Meaning
$\rho$	Weyl vector: half-sum of positive roots
$\ell(w)$	Length of Weyl group element w (number of positive roots it makes negative)
$\alpha^\vee$	Coroot: 2α/⟨α,α⟩
$A_\mu$	Alternating sum Σ_{w∈W} (−1)^{ℓ(w)} e^{w(μ)}

Properties

Character determines representation

If χ_λ = χ_μ then V_λ ≅ V_μ (characters distinguish irreducibles).

Weight multiplicity from character

The multiplicity of weight μ in V_λ is the coefficient of e^μ in the character χ_λ.

Kostant multiplicity formula

The multiplicity of μ in V_λ equals Σ_{w∈W} (−1)^{ℓ(w)} P(λ+ρ−w(μ+ρ)), where P is the Kostant partition function.

Theorems

Theorem 1: Weyl Character Formula

For a dominant integral weight λ: χ_λ = A_{λ+ρ}/A_ρ, where A_μ = Σ_{w∈W} (−1)^{ℓ(w)} e^{w(μ)}.

Theorem 2: Weyl Denominator Identity

A_ρ = Σ_{w∈W} (−1)^{ℓ(w)} e^{w(ρ)} = ∏_{α∈Φ⁺} (e^{α/2} − e^{−α/2}). This is a combinatorial identity relating the alternating Weyl group sum to a product over positive roots.

Theorem 3: Weyl Dimension Formula

dim V_λ = ∏_{α∈Φ⁺} ⟨λ+ρ, α∨⟩/⟨ρ, α∨⟩, obtained by setting e^{tμ} → 1+t⟨μ,·⟩ and taking the limit as t → 0.

Worked Examples

1
For sl(2): Φ⁺ = {α}, W = {1, s_α} with ℓ(s_α)=1. ρ = α/2. Dominant weights: λ = nα/2 for n ≥ 0.
$W = \{1, s_\alpha\}, \quad \rho = \alpha/2, \quad \lambda = n\alpha/2$
2
A_{λ+ρ} = e^{(n+1)α/2} − e^{−(n+1)α/2}. Weyl denominator A_ρ = e^{α/2} − e^{−α/2}.
$A_{\lambda+\rho} = e^{(n+1)\alpha/2} - e^{-(n+1)\alpha/2}, \quad A_\rho = e^{\alpha/2} - e^{-\alpha/2}$
3
χ_n = A_{n+1)/2} / A_{1/2} = (e^{(n+1)α/2} − e^{−(n+1)α/2})/(e^{α/2} − e^{−α/2}). Setting t = e^{α/2}: χ_n = (t^{n+1} − t^{−(n+1)})/(t − t^{−1}) = t^n + t^{n−2} + … + t^{−n}.
$\chi_n = \frac{e^{(n+1)\alpha/2} - e^{-(n+1)\alpha/2}}{e^{\alpha/2} - e^{-\alpha/2}} = e^{n\alpha/2} + e^{(n-2)\alpha/2} + \cdots + e^{-n\alpha/2}$
4
This is the sum of characters of weights n, n−2, …, −n, each with multiplicity 1, consistent with V_n having dim n+1.
$\dim V_n = n+1 \quad \checkmark$

✓ Answer

The Weyl character formula gives χ_n = e^{nα/2} + e^{(n−2)α/2} + ⋯ + e^{−nα/2}, correctly encoding the n+1 weight spaces of V_n.

Practice Problems

Hardfree response

Compute dim V_{(2,0)} for sl(3,ℂ) using the Weyl dimension formula.

Hardfree response

Write out the Weyl denominator A_ρ for sl(3,ℂ) as a product of factors (e^{α/2} − e^{−α/2}).

Quiz

Summary

Weyl character formula: χ_λ = A_{λ+ρ}/A_ρ, where A_μ = Σ_{w∈W} (−1)^{ℓ(w)} e^{w(μ)}.
The Weyl denominator A_ρ equals ∏_{α∈Φ⁺} (e^{α/2}−e^{−α/2}).
The Weyl dimension formula dim V_λ = ∏_{α>0} ⟨λ+ρ, α∨⟩/⟨ρ, α∨⟩ follows as a limit.
For sl(2), the formula recovers χ_n = e^{nα/2} + ⋯ + e^{−nα/2}.
The character encodes all weight multiplicities of V_λ.

References

BookHumphreys, J.E. — Introduction to Lie Algebras and Representation Theory (1972), Chapter 24
BookFulton, W. & Harris, J. — Representation Theory (1991), Appendix E