# The groups O(3), SO(3) and SU(2)

1. Jan 1, 2010

### Rory9

1. The problem statement, all variables and given/known data

How can irreducible representations of O(3) and SO(3) be determined from the irreducible representations of SU(2)?

3. The attempt at a solution

I believe there is a two-one homomorphic mapping from SU(2) to SO(3); is that enough for some shared representations? If I had an idea of *why* irreducible reps. can determined for O(3) and SO(3) from SU(2), I might have a better notion of *how* to go about proving it.

Cheers!

2. Jan 1, 2010

### George Jones

Staff Emeritus
Mathematically, what is a representation of a group G?

3. Jan 1, 2010

### Rory9

Typically a matrix, I believe, for which $$\Gamma(T_{1}T_{2}) = \Gamma(T_{1})\Gamma(T_{2})$$ holds, where $$T_{1}, T_{2}$$ belong to $$G$$

4. Jan 2, 2010

### dextercioby

There are 2 isomorphisms you need to use:

$$\mbox{SO(3)}\simeq\frac{\mbox{SU(2)}}{\mathbb{Z}_{2}}$$

and

$$\mbox{O(3)} = \mbox{SO(3)} \times \{-1_{3\times 3}, 1_{3\times 3} \}$$

There are at least 50 books or so discussing the connection b/w SO(3) and SU(2), however there are many less computing all representations of O(3) starting from the ones of SO(3) deducted from the ones of SU(2).

Last edited: Jan 2, 2010
5. Jan 2, 2010

### Rory9

Thank you very much for your answer. I understand the second statement, but what exactly are you doing in the first - simply slicing off the complex aspect by mathematical fiat?

Cheers :)