Given a representation R from SU(3) to G L V, we have seen that V equals the direct sum of weight spaces W_(k l) where k and l are integers and W_(k l) is the set of v in V such that R of the diagonal matrix e to the i theta_1, e to the i theta_2, e to the minus i (theta_1 plus theta_2) applied to v equals e to the i (k theta_1 plus l theta_2) times v or equivalently W_(k l) is the set of v in V such that R star superscript C of the diagonal matrix theta_1, theta_2, minus theta_1 minus theta_2 applied to v equals (k theta_1 plus l theta_2) times v.
Remember that the diagonal matrix theta_1, theta_2, minus theta_1 minus theta_2 isn't in little s u 3, rather it's in little s l 3 C, the complexification of little s u 3, which is why we're using R star superscript C.
We were drawing the weights (k, l) on a triangular lattice. For example, the weight diagram for the adjoint representation was:
We will change notation slightly and write W_(k l) equals W_(lambda) where lambda of theta equals k theta_1 plus l theta_2. Bundling the two integers together in this way will make life easier in future (e.g. when we have more than two integer weights).
Define L_1 of theta to be theta_1, L_2 of theta to be theta_2, L_3 of theta to be theta_3, which is minus theta_1 minus theta_2. These are the lambdas corresponding to (k, l) equals (1, 0), (0, 1), and (minus 1, minus 1) respectively.
With this notation, the weights of the standard representation are L_1, L_2, L_3 and the weights of the adjoint representation are L_i - L_j because little ad H_(theta) E_{i j} equals (theta_i minus theta_j) times E_{i j}.
For little s l 2 C, the adjoint representation has weight spaces W_(minus 2) spanned by Y, W_0 spanned by H and W_2 spanned by X. The elements X and Y played an important role in studying the representations of SU(2): X moved vectors from weight spaces with weight k to weight spaces with weight k + 2 and Y moved them back again.
The analogue for SU(3) will be to see how the weight vectors E_{i j} in little s l 3 C of the adjoint representation act on the weight spaces of another representation.
Given a complex representation R from S U 3 to G L V, R star superscript C of E_{i j} sends W_lambda to W_(lambda plus L_i minus L_j).
We illustrate the lemma in the figures below, showing how the matrices R star superscript C of E_{i j} act in the adjoint representation. For example R star superscript C of E_{1 3} and R star superscript C of E_{3 1} translate weight spaces forwards and backwards along the L_1 - L_3 direction.
The figure below shows the standard representation. There are three weights L_1, L_2, L_3. Let's see how E_{1 3}= 0, 0, 1; 0, 0, 0; 0, 0, 0 acts. It sends e_1 in W_{L_1} and e_2 in W_{L_2} to zero and it sends e_3 in W_{L_3} to e_1 in W_{L_1}. Correspondingly, we draw an arrow in the L_3 - L_1-direction in the weight diagram, as dictated by the lemma.
We know that E_{1 3} sends W_{L_1} to W_{2 L_1 - L_3} by the lemma, but W_{2 L_1 - L_3} = 0 which is why E_{1 3} e_1 = 0. In terms of the figure, the vector L_1 - L_3 starting at L_1 ends at a lattice point which is not in the weight diagram.
If v is in W_(lambda) then we need to show R star superscript C of E_{i j} applied to v is in W_{lambda plus L_i minus L_j}.
We have v in W_lambda if and only if R star superscript C of H_theta applied to v equals lambda of theta times v.
We have R star superscript C of E_{i j} applied to v is in W_{lambda plus L_i minus L_j} if and only if R star superscript C of H_theta times R star superscript C E_{i j} applied to v equals (lambda of theta plus thet_i minus theta_j) times R star superscript C of E_{i j} applied to v.
We have H_theta bracket E_{i j} equals little ad H_theta applied to E_{i j}, which equals (theta_i minus theta_j) times E_{i j}, because E_{i j} is in the L_i minus L_j weight space of the adjoint representation. Applying R star superscript C we get R star superscript C of (H_theta bracket E_{i j} equals R star superscript C of H theta bracket with R star superscript C of E_{i j}, which equals theta_i minus theta_j times R star superscript C of E_{i j}.
Therefore R star superscript C of H theta, times R star superscript C of E_{i j}, applied to v equals R star superscript C of E_{i j}, times R star superscript C of H_theta applied to v; plus (theta_i minus theta_j) times R star superscript C of E_{i j} applied to v.
Since v is in W_lambda, we have R star superscript C of H_theta applied to v equals lambda of theta times v, so R star superscript C of H_theta, times R star superscript C of E_{i j}, applied to v equals (lambda of theta plus theta_i minus theta_j) times R star superscript C of E_{i j} applied to v This shows that R star superscript C of E_{i j} applied to v is in W_(lambda plus L_i minus L_j) as required.
We have used two crucial things:
R star superscript C is a representation
little ad of H theta applied to E_{i j} equals (theta_i minus theta_j) times E_{i j}, in other words, E_{i j} is a root vector with root L_i minus L_j.
The same proof shows more generally that if X in little g is a weight vector of the adjoint representation (root vector) with weight (root) alpha then X sends weight vectors in W_lambda (for any representation) to weight vectors in W_(lambda plus alpha).
What do you think the weight diagram of the standard 4-dimensional representation of SU(4) would look like? How do you think the matrices E_{i j} act?