Every finite complex representation of a compact group is equivalent to a unitary representation

Let $𝐺$ be a compact group, $𝔛 : 𝐺 \to G L (𝑉)$ be a representation carried by a finite-dimensional complex inner product space $𝑉$ . Then $𝔛$ is equivalent to a unitary representation. #m/thm/rep2 Alternatively, there always exists an inner product on $𝑉$ for which $𝑉$ is unitary.¹

Proof

Let $𝜇$ be the normalised Haar measure for $𝐺$ . We define

(𝑣 | 𝑤) = \int 𝐺 ⟨ 𝔛 (𝑔) 𝑣 | 𝔛 (𝑔) 𝑤 ⟩ 𝑑 𝜇 (𝑔)

which is also an inner product on $𝑉$ since

conjugate symmetry

(𝑣 | 𝑤) = \int 𝐺 ⟨ 𝔛 (𝑔) 𝑣 | 𝔛 (𝑔) 𝑤 ⟩ 𝑑 𝜇 (𝑔) = \int 𝐺 ―――――――― ⟨ 𝔛 (𝑔) 𝑤 | 𝔛 (𝑔) 𝑣 ⟩ 𝑑 𝜇 (𝑔) = ―――― (𝑤 | 𝑣)

linear in second argument

(𝑣 | 𝛼 𝑤 + 𝛽 𝑢) = \int 𝐺 ⟨ 𝔛 (𝑔) 𝑣 | 𝛼 𝔛 (𝑔) 𝑤 + 𝛽 𝔛 (𝑔) 𝑢 ⟩ 𝑑 𝜇 (𝑔) = 𝛼 \int 𝐺 ⟨ 𝔛 (𝑔) 𝑣 | 𝔛 (𝑔) 𝑤 ⟩ 𝑑 𝜇 (𝑔) + 𝛽 \int 𝐺 ⟨ 𝔛 (𝑔) 𝑣 | 𝔛 (𝑔) 𝑢 ⟩ 𝑑 𝜇 (𝑔) = 𝛼 (𝑣 | 𝑤) + 𝛽 (𝑣 | 𝑢)

positive definite

(𝑣 | 𝑣) = \int 𝐺 ⟨ 𝔛 (𝑔) 𝑣 | 𝔛 (𝑔) 𝑣 ⟩ ⏟___⏟___⏟ > 0 𝑑 𝜇 (𝑔) > 0

Let ${𝑣 𝑗}$ be an Orthonormal basis with respect to $⟨ \cdot | \cdot ⟩$ and ${𝑤 𝑗}$ be an orthonormal basis with respect to $(\cdot | \cdot)$ . Then there exists an invertible change of basis $𝑆 : 𝑉 \to 𝑉$ with $𝑆 𝑤 𝑗 = 𝑣 𝑗$ , which is also a Change of inner product with $(𝑣 | 𝑤) = ⟨ 𝑆 𝑣 | 𝑆 𝑤 ⟩$ . We define

˜ 𝔛 (𝑔) = 𝑆 𝔛 (𝑔) 𝑆 - 1

which is equivalent to $𝔛$ , and unitary since

⟨ ˜ 𝔛 (𝑔) 𝑣 | ˜ 𝔛 (𝑔) 𝑤 ⟩ = ⟨ 𝑆 𝔛 (𝑔) 𝑆 - 1 𝑣 | 𝑆 𝔛 (𝑔) 𝑆 - 1 𝑤 ⟩ = (𝔛 (𝑔) 𝑆 - 1 𝑣 | 𝔛 (𝑔) 𝑆 - 1 𝑤) = \int 𝐺 ⟨ 𝔛 (ℎ 𝑔) 𝑆 - 1 𝑣 | 𝔛 (ℎ 𝑔) 𝑆 - 1 𝑤 ⟩ 𝑑 𝜇 (ℎ) = \int 𝐺 ⟨ 𝔛 (ℎ) 𝑆 - 1 𝑣 | 𝔛 (ℎ) 𝑆 - 1 𝑤 ⟩ 𝑑 𝜇 (ℎ) = (𝑆 - 1 𝑣 | 𝑆 - 1 𝑤) = ⟨ 𝑣 | 𝑤 ⟩

as required.²

Infinite, non–compact groups

A simple counterexample to this result for a nonfinite group may be achieved with $𝔛 : ℤ \to G L (ℂ) : 𝑛 \mapsto 𝑎 𝑛$ where $𝑛 \in ℂ ∖ {0}$ . For $𝑎 \neq 1$ the representation is not unitary under the only inner product $ℂ$ supports $⟨ 𝑧 | 𝑤 ⟩ = 𝑧 * 𝑤$ .¹

#state/tidy | #lang/en | #SemBr

1996, Representations of finite and compact groups, pp. 21–22 ↩ ↩²
2021, Groups and representations, pp. 21–22 ↩

Every finite complex representation of a compact group is equivalent to a unitary representation

Infinite, non–compact groups

Footnotes