Every finite complex representation of a compact group is equivalent to a unitary representation
Let
Proof
Let
be the normalised Haar measure for π . We define πΊ ( π£ | π€ ) = β« πΊ β¨ π ( π ) π£ | π ( π ) π€ β© π π ( π ) which is also an inner product on
since π
- conjugate symmetry
( π£ | π€ ) = β« πΊ β¨ π ( π ) π£ | π ( π ) π€ β© π π ( π ) = β« πΊ βββββββ β¨ π ( π ) π€ | π ( π ) π£ β© π π ( π ) = ββββ ( π€ | π£ )
- linear in second argument
( π£ | πΌ π€ + π½ π’ ) = β« πΊ β¨ π ( π ) π£ | πΌ π ( π ) π€ + π½ π ( π ) π’ β© π π ( π ) = πΌ β« πΊ β¨ π ( π ) π£ | π ( π ) π€ β© π π ( π ) + π½ β« πΊ β¨ π ( π ) π£ | π ( π ) π’ β© π π ( π ) = πΌ ( π£ | π€ ) + π½ ( π£ | π’ )
- positive definite
( π£ | π£ ) = β« πΊ β¨ π ( π ) π£ | π ( π ) π£ β© β __ β __ β > 0 π π ( π ) > 0 Let
be an Orthonormal basis with respect to { π£ π } and β¨ β | β β© be an orthonormal basis with respect to { π€ π } . Then there exists an invertible change of basis ( β | β ) with π : π β π , which is also a Change of inner product with π π€ π = π£ π . We define ( π£ | π€ ) = β¨ π π£ | π π€ β© Λ π ( π ) = π π ( π ) π β 1 which is equivalent to
, and unitary since π β¨ Λ π ( π ) π£ | Λ π ( π ) π€ β© = β¨ π π ( π ) π β 1 π£ | π π ( π ) π β 1 π€ β© = ( π ( π ) π β 1 π£ | π ( π ) π β 1 π€ ) = β« πΊ β¨ π ( β π ) π β 1 π£ | π ( β π ) π β 1 π€ β© π π ( β ) = β« πΊ β¨ π ( β ) π β 1 π£ | π ( β ) π β 1 π€ β© π π ( β ) = ( π β 1 π£ | π β 1 π€ ) = β¨ π£ | π€ β© as required.2
Infinite, nonβcompact groups
A simple counterexample to this result for a nonfinite group may be achieved with
Footnotes
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1996, Representations of finite and compact groups, pp. 21β22 β© β©2
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2021, Groups and representations, pp. 21β22 β©