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.

source
jl@crush.caltech.edu index