A representation is called unitary <=> D(g) is unitary forall g in G.
Properties:
I will use ` to denote "dagger"= transpose and conjugate
* D(g)` D(g) = e
* D1(g) not unitary, and D2(g) unitary with D1(g)=M^D2(g)M can occur.
Is D(g) equivalent to a unitary representation?
* G finite => yes
* G infinit => maybe.
Theorem: given N dim representation, D, of a finite group G of order h,
D is equivalent to a unitary representation of G.
Proof:
Let X = Sum(i=1,h,D(g)`D(g)). X is hermetian and positive definite.
there exists a unitary matrix, U, s.t.
X=U^(Diag)U where Diag =
|d1 |
| . |
| . |
| dn|
with di's = eigenvalues of X.
X positive definite => di>0 forall i.
X=S*S with S=U^*(Diag^1/2)*U with S hermetian where (Diag^1/2) =
|d1^.5 |
| . |
| . |
| dn^.5|
forall a in G: D`(a)XD(a)=Sum(i=1,h,D`(a)D`(gi)D(gi)D(a))=Sum(i=1,h,D`(gia)D(gia))=X
D`S^2D=S^2
let D~=SDS^ => D~`=(SDS^)`=S^`D`S`=S^D`S
D~`D~=I => D~ is unitary and D~ is equivalent to D.
Theorem: any non-compact group has no finite dime unitary irreps.