Unitary representation

A complex representation is unitary when every representing matrix preserves a positive-definite Hermitian inner product.

A unitary irreducible representation is both unitary and irreducible. Such representations are the building blocks in the harmonic analysis of groups and in quantum symmetry classifications.

Unitarization of a finite-group representation

Average a positive-definite Hermitian form over a finite group and choose an orthonormal basis for the averaged form. In that basis every representing matrix is unitary.

For a compact group with normalized Haar measure, averaging any positive-definite Hermitian form,

⟨ v, w ⟩_{G} = \int_{G} ⟨ ρ (g) v, ρ (g) w ⟩ d g,

produces an invariant positive-definite Hermitian form.

ρ : G \to G L (V)

of a compact group, average any positive-definite real inner product against normalized Haar measure:

(v, w)_{G} = \int_{G} (ρ (g) v, ρ (g) w) d g .

The result is positive definite and

G

-invariant. An orthonormal basis for it conjugates the representation into an orthogonal group.

S^{1}

is unitary and decomposes into one-dimensional weight spaces. Its continuous one-dimensional characters are

z \mapsto z^{m}

for

m \in Z

If a central subgroup is isomorphic to

S^{1}

, its weight spaces are invariant under the entire group, because every representing operator commutes with the central circle action.

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