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 $C\_n$.



Tensor product of fundamental representations
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 $\mathfrak{g}$.


