Highest weights of irreducible components of tensor product of irreducible sl(3)-module [closed]

Question

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows:

For each weight $\mu$, let $L(\mu)$ be the irreducible $sl(3)$-module of highest weight $\mu$ and assume that $L(\alpha)$ and $L(\beta)$ are finite dimensional.

Is it true that the tensor representation $L(\alpha) \otimes L(\beta) =\bigoplus_{i=1}^n L(\lambda_i) $ satisfying that for each $\lambda_i$ $\lambda_i = \alpha + \gamma$ for some weight $\gamma$ of the module $L(\beta)$?

I have computed several cases, this result hold in those cases.

Thank you very much !

Answer 1

I'm assuming that you meant "is it true... satisfies that". In which case the answer is yes, not just for $\mathfrak{sl}(3)$ but any semisimple Lie algebra. One of the many ways to see this uses the Littelmann path model, in which the highest weight paths in the tensor product crystal are (only some of the) concatenations of the highest weight path for $L(\alpha)$ and a path for $L(\beta)$.

I don't view this as a research-level question. Also, please don't use $\alpha$ to denote a dominant weight, because everybody uses it for a simple root.