It's important to note that this is not a decomposition of as a representation of , only as a representation of . In particular, if you act with a non-diagonal matrix in then it will mix up these subspaces .
We now embark on our study of the fine structure of representations, aiming towards the classification theorem for irreducible representations that we stated earlier.
The first key observation is that contains a subgroup isomorphic to :
If I have a complex representation then I can restrict it to and I get a representation . By the classification of irreps of , we find that where and, with respect to this splitting, is the diagonal matrix . The are integers called the weights, the are called weight spaces, and the direct sum is called the weight space decomposition.
Take the standard representation . We have . This is already diagonal. The weight spaces are spanned by the vectors and and the weights are and respectively.
We will depict the weight space decomposition as follows (the weight diagram). Consider the integer points sitting on the number line. Colour in each integer which appears as a weight, and label it by the number of having weight if this number is strictly bigger than 1. In the example above, the weight diagram is a blob at and a blob at .
Consider the representation , . We computed the associated Lie algebra homomorphism . We found that Since we're interested in the diagonal subgroup we can set . We're interested in which gives The weights are going to be . This is because:
We calculated the weights of the representation , : these were because with respect to a basis of eigenvectors this matrix diagonalised and became .
We're looking at the same submatrix but with everywhere instead of , so this submatrix can be diagonalised to give .
What's left is the top-left entry which is , so we also get a weight space with weight 0.
The weight diagram is therefore:
Take . If and form the standard basis for the standard representation then this has basis , , . Since acts as in the standard representation, acts as (each factor transforms under the standard representation and then you multiply them together and the factors of or either combine or cancel).
The weight spaces are therefore:
, with weight ,
, with weight ,
, with weight .
The weight diagram is therefore the same as the weight diagram for the previous example:
We'll see later that this is enough to tell us that these representations are isomorphic.
Figure out the weight diagram for .