Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $V_1,\cdots, V_n$ be the fundamental representations (the irreducible ones with fundamental weights $\omega_1,\cdots,\omega_n$). Take a $k$-tensor product of these representations: $V_{\lambda_1}\otimes\cdots\otimes V_{\lambda_k}$ (with each $\lambda_i\in\{\omega_1,\cdots,\omega_n\}$).
Decompose this product into irreducible representations. Let $\lambda=\sum n_i\omega_i$ be the highest weight of such a simple summand. Can we conclude $\sum n_i\leq k$?
I can show it holds for type $A_n$ and $C_n$.
Best Answer
Actually one can show that if $\sum n_i \lambda_i$ is a highest weight in a $k$-tensor product of fundamental representations, we have $\sum n_i\leq \beta \cdot k$ for some $\beta$ uniquely determined by the simple type of $\mathfrak{g}$.