Group representation

Unitary representation

A unitary representation $𝔛$ of $𝐺$ is a group homomorphism $𝔛 : 𝐺 \to U (𝑉)$ into the unitary group #m/def/rep2 where $𝑉$ is a complex inner product space. Thus $𝔛$ is a group representation carried by $𝑉$ such that

⟨ 𝔛 (𝑔) 𝑣 | 𝔛 (𝑔) 𝑤 ⟩ = ⟨ 𝑣 | 𝑤 ⟩

for all $𝑔 \in 𝐺$ and $𝑣, 𝑤 \in 𝑉$ where $⟨ - | - ⟩$ is the inner product.

Properties

Every finite complex representation of a compact group is equivalent to a unitary representation.
The orthogonal complement of an invariant subspace under a unitary operator is invariant

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