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I am wondering about a combinatorial formula for computing the weights of the irreducible representations $\Gamma_{a,b}$ of $\mathfrak{sl}_3\Bbb C$. By $\Gamma_{a,b}$ I mean the irrep that has highest weight $aL_1 - bL_3$, which may be thought of concretely as $\mathrm{Sym}^a V \otimes \mathrm{Sym}^b V^\vee$, where $V \cong \Bbb C^3$ denotes the standard representation. Would it involve the Schur polynomials?

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If you write your highest weight in terms of the fundamental weights, then they will be of the form $a_1 \omega_1 + a_2 \omega_2$, where $\omega_1, \omega_2$ are the two fundamental weights of $\mathfrak{sl}_3$. Then the dimension of the weight space of weight $b_1 \epsilon_1 + b_2 \epsilon_2 + b_3 \epsilon_3$ is the number of semistandard tableaux with $a_2$ columns of length 2, $a_1$ columns of length 1 and with the number of boxes containing the entry $i$ equal to $b_i$ for $i=1,2,3$. If you are interested only in which weights occur, and not in the dimensions of the weight spaces, then you only need to determine whether or not a filling with a specific number of boxes of each type occurs.

This can all be generalized naturally to $\mathfrak{sl}_n$ or $\mathfrak{gl}_n$.

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