Unitary representation

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

Let be a compact group, be a representation carried by a finite-dimensional complex inner product space . Then is equivalent to a unitary representation. rep2 Alternatively, there always exists an inner product on for which is unitary.¹

Proof

Let be the normalised Haar measure for . We define

which is also an inner product on since

1. conjugate symmetry

2. linear in second argument

3. positive definite

Let be an Orthonormal basis with respect to and be an orthonormal basis with respect to . Then there exists an invertible change of basis with , which is also a Change of inner product with . We define

which is equivalent to , and unitary since

as required.²

Infinite, non–compact groups

A simple counterexample to this result for a nonfinite group may be achieved with where . For the representation is not unitary under the only inner product supports .¹

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Footnotes

1. 1996, Representations of finite and compact groups, pp. 21–22 ↩ ↩²

2. 2021, Groups and representations, pp. 21–22 ↩