Highest Weight Theory — Mathematics Knowledge System

Overview

Highest weight theory provides a complete classification of finite-dimensional irreducible representations of complex semisimple Lie algebras. Every such representation is uniquely determined by a dominant integral weight — a linear functional on the Cartan subalgebra satisfying an integrality and positivity condition. This remarkable rigidity means the entire representation theory is organised by a lattice of weights, with the Weyl character formula giving the characters of all irreducible representations.

Intuition

In a representation of a semisimple Lie algebra, the Cartan subalgebra acts diagonally — splitting the representation space into weight spaces. The 'raising' operators (positive root spaces) move between weight spaces, and there must be a top weight vector that cannot be raised further. This highest weight uniquely determines the entire representation, much as the highest eigenvalue of a symmetric matrix encodes special information.

Formal Definition

Definition

Let 𝔤 be a semisimple Lie algebra with Cartan subalgebra 𝔥, root system Φ, and chosen positive roots Φ⁺. A weight of a representation (V, ρ) is a functional λ ∈ 𝔥* such that the weight space Vλ = {v ∈ V : ρ(H)v = λ(H)v ∀H ∈ 𝔥} is nonzero.

V = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda, \quad V_\lambda = \{v \in V : H \cdot v = \lambda(H)v \; \forall H \in \mathfrak{h}\}

Weight space decomposition

\lambda \in \mathfrak{h}^* \text{ is dominant integral} \iff \langle \lambda, \alpha^\vee \rangle \in \mathbb{Z}_{\ge 0} \; \forall \alpha \in \Phi^+

Dominant integral weight condition

v_\lambda \in V_\lambda \text{ is a highest weight vector} \iff e_\alpha \cdot v_\lambda = 0 \; \forall \alpha \in \Phi^+

Highest weight vector annihilated by all raising operators

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

Weyl character formula

Notation

Notation	Meaning
$\lambda$	Weight — element of 𝔥*
$\Lambda^+$	Set of dominant integral weights
$L(\lambda)$	Irreducible representation with highest weight λ
$\rho = \frac{1}{2}\sum_{\alpha \in \Phi^+} \alpha$	Weyl vector (half-sum of positive roots)
$W$	Weyl group
$\ell(w)$	Length of Weyl group element w

Properties

Classification theorem

\text{The map } \lambda \mapsto L(\lambda) \text{ is a bijection from } \Lambda^+ \text{ to isomorphism classes of finite-dimensional irreducible representations of } \mathfrak{g}

Dimension formula (Weyl)

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

Multiplicity of weights

\dim V_\mu \le \dim V_{w(\mu)} \text{ and } \dim V_{w(\lambda)} = \dim V_\lambda \; \forall w \in W

Worked Examples

1
sl₂(ℂ) has standard 2D representation V = ℂ² with basis v₁, v₂.
$V = \mathbb{C}^2$
2
h acts as diag(1,-1), so h·v₁ = v₁ and h·v₂ = -v₂.
$h \cdot v_1 = v_1,\quad h \cdot v_2 = -v_2$
3
Weights are λ₁ = 1 (weight space ℂv₁) and λ₂ = -1 (weight space ℂv₂).
$V_1 = \mathbb{C}v_1,\quad V_{-1} = \mathbb{C}v_2$
4
Highest weight is λ = 1; this is L(1), the standard representation.
$L(1) \cong \mathbb{C}^2$

✓ Answer

The standard representation has weights ±1, highest weight 1, and is the unique 2-dimensional irreducible.

Practice Problems

Hardfree response

Show that the n-th symmetric power Sⁿ(ℂ²) is an irreducible sl₂(ℂ)-representation with highest weight n.

Common Mistakes

Common Mistake

Confusing weights and roots

Roots are weights of the adjoint representation. General weights of an arbitrary representation are elements of 𝔥* that need not be roots.

Common Mistake

Assuming every weight appears with multiplicity 1

In general, weights can have multiplicity greater than 1. The highest weight always has multiplicity 1, but other weights may not.

Quiz

Which of the following is a dominant integral weight for sl₂(ℂ)?

λ = -1λ = 1/2λ = 3λ = -1/2

The Weyl character formula expresses:

The dimension of weight spacesThe character as an alternating sum over the Weyl groupThe decomposition of tensor productsThe index of the representation

Historical Background

The theory was pioneered by Élie Cartan in his 1913 paper on irreducible representations, and reformulated algebraically by Hermann Weyl in the 1920s via his character formula. The Borel–Weil theorem (1954) gave a geometric construction of all representations via line bundles on flag varieties, opening the door to geometric representation theory.

1913
Cartan classifies irreducible representations by highest weights
Élie Cartan
1925
Weyl's character formula expresses characters of irreducible representations
Hermann Weyl
1954
Borel–Weil theorem realises representations geometrically
Armand Borel, André Weil

Summary

Every finite-dimensional irreducible representation of a semisimple Lie algebra is determined up to isomorphism by its highest weight λ ∈ Λ⁺.
Dominant integral weights are those with ⟨λ, α∨⟩ ∈ ℤ≥0 for all positive roots α.
The Weyl character formula computes the character of L(λ) as an alternating sum over the Weyl group.
The Weyl dimension formula gives dim L(λ) = ∏_{α∈Φ⁺} ⟨λ+ρ, α∨⟩/⟨ρ, α∨⟩.

References

BookHumphreys, J.E. — Representations of Semisimple Lie Algebras in the BGG Category O (2008)
BookHumphreys, J.E. — Introduction to Lie Algebras and Representation Theory (1972), Chapter VI
WebsiteWikipedia — Weight (representation theory)
WebsitenLab — highest weight representation