Representations of Compact Lie Groups — Mathematics Knowledge System

Overview

Compact Lie groups — such as U(n), SU(n), SO(n), and Sp(n) — have a completely developed and beautiful representation theory. Every representation is unitarisable (Haar-measure averaging), every representation is completely reducible, and the irreducible representations are classified by their highest weights via the Cartan–Weyl theory. The Peter–Weyl theorem gives an L²-decomposition, and the Weyl character formula computes characters of irreducibles.

Intuition

Compact Lie groups interpolate between finite groups (which have finitely many irreps) and non-compact groups (which can have wild representations). A compact Lie group has countably many irreducible representations, each finite-dimensional, and the structure is as clean as for finite groups — averaging over the Haar measure plays the role of summing over a finite group.

Formal Definition

Definition

A compact Lie group G has a unique (up to normalisation) bi-invariant Haar measure. A representation on a finite-dimensional complex vector space V is a smooth group homomorphism ρ: G → GL(V).

\text{Every finite-dim rep is completely reducible: } V \cong \bigoplus_i V_i \quad (V_i \text{ irreducible})

Complete reducibility via Haar-measure averaging

\text{Irred. reps classified by dominant weights: } \lambda \in \Lambda^+ = \{\sum_i n_i \omega_i : n_i \in \mathbb{Z}_{\geq 0}\}

Highest weight classification

\chi_{\lambda}(g) = \frac{\sum_{w \in W} (-1)^{\ell(w)} e^{w(\lambda+\rho)-\rho}}{\prod_{\alpha>0}(e^{\alpha/2}-e^{-\alpha/2})}

Weyl character formula (schematic)

\dim V_\lambda = \prod_{\alpha > 0} \frac{\langle \lambda + \rho, \alpha \rangle}{\langle \rho, \alpha \rangle}

Weyl dimension formula

Notation

Notation	Meaning
$\Lambda^+$	Dominant integral weights
$\omega_i$	Fundamental weights
$W$	Weyl group of G
$\rho$	Half-sum of positive roots (Weyl vector)

Properties

Schur orthogonality

⟨χ_λ, χ_μ⟩_G = δ_{λμ} for irreducible characters χ_λ, χ_μ.

Tensor product decomposition

V_λ ⊗ V_μ ≅ ⊕_{ν} m^ν_{λμ} V_ν, where the multiplicities m^ν_{λμ} are the Littlewood–Richardson coefficients (for GL_n) or computed via the Weyl character formula in general.

Theorems

Theorem 1: Complete Reducibility

Every finite-dimensional continuous representation of a compact Lie group is completely reducible: it decomposes as a finite direct sum of irreducible representations.

Theorem 2: Cartan–Weyl Highest Weight Theorem

The irreducible finite-dimensional representations of a compact Lie group G are in bijection with the dominant integral weights Λ⁺. Each dominant weight λ corresponds to a unique (up to isomorphism) irreducible representation V_λ.

Theorem 3: Weyl Integration Formula

For a class function f on G: ∫_G f(g)dg = (1/|W|) ∫_T f(t)|Δ(t)|² dt, where T is a maximal torus, W the Weyl group, and Δ(t) = ∏_{α>0}(e^{α/2}(t)−e^{−α/2}(t)) the Weyl denominator.

Worked Examples

1
U(1) = {e^{iθ}} is abelian. Its irreducible representations are all 1-dimensional: ρₙ(e^{iθ}) = e^{inθ} for n ∈ ℤ.
$\hat{U}(1) = \{\rho_n : n \in \mathbb{Z}\}$
2
The Lie algebra is iℝ with generator h = i. The weight of ρₙ is n ∈ ℤ. The dominant weights are n ≥ 0 (with ρ = 0 since U(1) has no positive roots).
$\Lambda^+ = \mathbb{Z}_{\geq 0} \quad (\text{for } U(1))$
3
For negative n, ρₙ is the dual of ρ_{|n|}. The full set ℤ of weights indexes all irreducibles (including duals).
$\rho_{-n} = \rho_n^* \quad (\text{contragredient})$

✓ Answer

Irreducibles of U(1) are {e^{inθ}} for n ∈ ℤ, classified by weight n. Dominant weights are n ≥ 0.

Practice Problems

Mediumfree response

State why every representation of a compact Lie group is completely reducible, citing the key ingredient.

Hardfree response

What is the Weyl dimension formula for SU(3) applied to the adjoint representation (highest weight (1,1) in terms of fundamental weights)?

Quiz

Summary

Every representation of a compact Lie group is unitarisable and completely reducible.
Irreducible representations are classified by dominant integral weights via the Cartan–Weyl theorem.
The Weyl character formula computes characters; the Weyl dimension formula gives dim V_λ.
Peter–Weyl: L²(G) decomposes as ⊕_λ V_λ ⊗ V_λ*.
The Weyl integration formula reduces G-integrals to torus integrals.

References

BookBröcker, T. & tom Dieck, T. — Representations of Compact Lie Groups (1985)
BookAdams, J.F. — Lectures on Lie Groups (1969)