For each finite irreducible representation of Lie algebra $su(3)$ one knows that it is characterized by highest weight $(\lambda_1, \lambda_2)$ with integral entries. In this notation, $(1,0)$ is fundamental representation, $(0,1)$ is anti-fundamental representation, and $(1,1)$ is adjoint representation.

Given three highest weight vectors, say $(\lambda_1, \lambda_2)$ and $(\mu_1, \mu_2)$, $(\nu_1,\nu_2)$, is there an arithmetic way to determine if $(\nu_1,\nu_2)$ is in the tensor product decomposition of $(\lambda_1, \lambda_2) \otimes (\mu_1, \mu_2)$?

The analogy of this is the tensor product decompostion of $su(2)$, where we know $j$ is in $j_1 \otimes j_2$ if $|j_1 - j_2| < j < j_1+j_2$. Is the same thing happen for $su(3)$ so that one does not need to go through Young tableau prescription?


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