Can anyone please walk me through how to use the diagram below to decompose $\mathbb{C^3} \otimes V_{1,1}$ as a direct sum of irreducible representations of Lie algebra $sl_3$. Here $V_{1,1}$ is the irreducible representation of $sl_3$ with highest weight $\omega_1 + \omega_2$
I have little understanding on how to use diagram so any explanations, hints and answers given will be utterly appreciated.
Follow these steps:
(i) Find the weights of the rep $\mathbb C^3$. Also find the weights of $V_{1,1}$.
(ii) Deduce the weights of the tensor rep $\mathbb C^3 \otimes V_{1,1}$. [You should be able to get these by adding the weights of $\mathbb C^3$ and $V_{1,1}$ in pairs.] Draw these weights on the diagram! Make sure you keep track of their multiplicities.
(iii) Identify the highest weight of $\mathbb C^3 \otimes V_{1,1}$. What are the weights of the irreducible representation whose highest weight is the same as the highest weight of $\mathbb C^3 \otimes V_{1,1}$?
(iv) Remove a copy of this irreducible representation from $\mathbb C^3 \otimes V_{1,1}$. Which weights of $\mathbb C^3 \otimes V_{1,1}$ still remain after you have removed this irreducible representation? What is the highest weight among the weights that still remain? What are the weights in the irreducible representation with this highest weight vector?
Repeat step (iv) until there are no weights remaining.
[A useful fact: The weights of the irreducible representation with a given highest weight all lie within the convex hull of the hexagon that has a corner equal to this highest weight. To be more precise, the weights that appear are those within the convex hull of this hexagon that can be reached by acting on the highest weight with integer combinations of the translations $-\omega_1 - 2\omega_2$ and $-2\omega_1 - \omega_2$. The weights in the "outer shell" of the hexagon each appear in this irreducible representation with multiplicity one. Moving inwards, shell by shell, the multiplicities increase by one on every successive shell, until you reach a shell that is triangular, from which point onwards the multiplicities remain the constant as you move inwards shell by shell. Fulton and Harris has a better explanation of this.
Thus the rep $V_{1,1}$ has eight weights altogether, counted with multiplicity. There are six weights in the "outer shell" of the hexagon, namely $\pm(\omega_1 + \omega_2), \pm(-\omega_1 + 2\omega_2), \pm (2\omega_1 - \omega_2)$, which each appear with multiplicity one. Then there is the weight $0$, at the centre, which appears with multiplicity two.]