§ Representation theory of [TODO ]
2x2 unitary matrices, so .
- Lie algebra is , which are of the form , and .
- We write .
- The group elements are matrices, so this is the standard representation, which goes from to . Turns out this is irreducible, 2D complex representation.
- We have a transformation which for a creates a map which sends a matrix to . so the representation is , which has type signature . This is a 3D, real representation: the vectors have 3 degrees of freedom.
- We like complex representations, so we're going to build .
- There is the trivial representation .
- There is a zero dimensional representation which maps to . So it's the identity transformation on .
For any integer there is an irrep . Also, any irrep is isomorphic to one of these.
§ New representations from old
- If we have and , what are new representations?
- For one, we can build the direct sum . But this is useless, since we don't get irreps.
- We shall choose to take tensor product of representations.
- Symmetric power of is . This is not irreducible because it contains a subrep of symmetric tensors .
- Example, in , we can consider , , and .
- Define (for averaging) of to be . In other words, it symmetrizes an input tensor.
- Define . We claim that is a suprep of . We do this by first showing that is a morphism of representations, and then by showing that the image of a morphism is a sub-representation.
§ Weight space decomposition
- contains a subgroup isomorphic to . Call this subgroup , which is of the form .