Unitary Representations — Mathematics Knowledge System

Overview

A unitary representation of a group G is a group homomorphism π: G → U(H) into the group of unitary operators on a Hilbert space H. Unitary representations are central to quantum mechanics, where the symmetry group of a physical system acts unitarily on the Hilbert space of states. Every continuous representation of a compact group on a Hilbert space is unitarisable, and the Peter–Weyl theorem gives a complete decomposition of L²(G).

Intuition

Unitary operators preserve inner products — they are the 'isometries' of Hilbert space. A unitary representation says the group acts by rigid motions, preserving lengths and angles. This is the natural setting for quantum symmetries and Fourier analysis on groups.

Formal Definition

Definition

A unitary representation of G on a Hilbert space H is a group homomorphism π: G → U(H) where U(H) denotes the group of unitary operators. For topological groups one additionally requires strong continuity: g ↦ π(g)v is continuous for each v ∈ H.

\pi : G \to U(H), \quad \langle \pi(g)v, \pi(g)w \rangle = \langle v, w \rangle \quad \forall g \in G,\; v, w \in H

Unitary representation: isometries of Hilbert space

\pi(g)^* = \pi(g)^{-1} = \pi(g^{-1})

Adjoint equals inverse (unitarity)

\text{Strongly continuous: } g \mapsto \pi(g)v \in H \text{ is continuous for all } v \in H

Strong continuity (for topological groups)

\mathrm{Hom}_G^{\mathrm{unit}}(H_1, H_2) = \{ T : H_1 \to H_2 \text{ bounded linear} \mid T\pi_1(g) = \pi_2(g)T \}

G-equivariant bounded maps (intertwiners)

Notation

Notation	Meaning
$U(H)$	Group of unitary operators on Hilbert space H
$\pi$	Unitary representation homomorphism
$\langle \cdot, \cdot \rangle$	Inner product on H
$\pi(g)^*$	Hilbert-space adjoint of π(g)

Properties

Dual is conjugate

For a unitary representation, the contragredient ρ* is isomorphic to the complex conjugate representation ρ̄.

Character is real on involutions

If g² = e then χ(g) = χ(g⁻¹) = χ̄(g), so χ(g) ∈ ℝ.

Theorems

Theorem 1: Unitarisability of Compact Group Representations

Every continuous finite-dimensional representation of a compact group G is unitarisable: there exists a G-invariant inner product, obtained by averaging any inner product over G with the Haar measure.

Theorem 2: Schur's Lemma (unitary version)

If π₁ and π₂ are irreducible unitary representations and T: H₁ → H₂ is a bounded intertwiner, then either T = 0 or T is a scalar multiple of a unitary isomorphism.

Theorem 3: Complete Reducibility

Every unitary representation of a group decomposes as a direct integral (or direct sum in the compact case) of irreducible unitary representations.

Worked Examples

1
The Hilbert space is H = ℓ²(G) with orthonormal basis {eₕ : h ∈ G}. The regular representation is (λ(g)f)(h) = f(g⁻¹h).
$(\lambda(g) f)(h) = f(g^{-1}h), \quad f \in \ell^2(G)$
2
Check unitarity: ⟨λ(g)f₁, λ(g)f₂⟩ = Σ_h f₁(g⁻¹h)·f₂(g⁻¹h)̄. Substituting k = g⁻¹h (which is a bijection G → G) gives Σ_k f₁(k)f₂(k)̄ = ⟨f₁, f₂⟩.
$\langle \lambda(g)f_1, \lambda(g)f_2 \rangle = \sum_{h \in G} f_1(g^{-1}h)\overline{f_2(g^{-1}h)} = \sum_{k \in G} f_1(k)\overline{f_2(k)} = \langle f_1, f_2 \rangle$

✓ Answer

Left translation preserves the inner product by the substitution k = g⁻¹h; thus λ is unitary.

Practice Problems

Mediumfree response

Let G = U(1) = {e^{iθ} : θ ∈ ℝ} and define πₙ(e^{iθ}) = e^{inθ} for n ∈ ℤ. Show this is a unitary representation on H = ℂ.

Mediumproof writing

Show that if π is an irreducible unitary representation of an abelian group G, then dim H = 1.

Quiz

Summary

A unitary representation π: G → U(H) preserves the inner product: ⟨π(g)u, π(g)v⟩ = ⟨u, v⟩.
For compact groups, every continuous representation is unitarisable via averaging over Haar measure.
Schur's lemma implies irreducible unitary representations of abelian groups are 1-dimensional.
The contragredient of a unitary representation equals its complex conjugate.
Unitary representations are completely reducible (as a direct integral/sum of irreducibles).

References

BookFolland, G.B. — A Course in Abstract Harmonic Analysis (1995), Chapter 3
BookKnapp, A.W. — Representation Theory of Semisimple Groups (1986), Chapter 1