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Let $\omega_k$ be the highest weight of the $k$-th antisymmetric representation of $\mathfrak{su}(N)$. Consider an irreducible representation of $\mathfrak{su}(N)$, characterized by its highest weight $\omega = \sum_{j=1}^{N-1} n_j \omega_j$ for some integers $n_j\geq 0$.

What is the decomposition of this representation into irreducible representations of $\mathfrak{su}(N-1)$? I suspect that there will be $\prod_{j=1}^{N-1} (n_j+1)$ pieces labelled by integers $0\leq k_j\leq n_j$, with highest weights $$ \sum_{j=1}^{N-1} \bigl(k_j \omega_j + (n_j - k_j) \omega_{j-1}\bigr) $$ Am I right?

I've obviously failed to find the answer by looking online, and will be very grateful if someone finds a reference for me to read.

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