Tensor product of fundamental representations

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 $$C_n$$.

• Can we not conclude immediately that $\sum n_i = k$ for the "highest highest" weight? The tensor product of $V_\lambda$ and $V_\mu$ contains $V_{\mu + \lambda}$ and all other irreducible subrepresentations have lower highest weight. Extending this to $V_{\lambda_1} \otimes \cdots \otimes V_{\lambda_k}$ we see that the highest weight present overall is $\lambda_1 + \cdots + \lambda_k$ (so all other highest weights are lower than this) from which the result you want follows. Nov 28, 2022 at 14:32
• That is true. But some lower weight of the form ($\sum n_i w_i -$ simple roots) may have larger $\sum n_i$. Nov 28, 2022 at 15:53
• Ah good point and indeed playing around with some examples produces a counter example: in $D_5$, $V_{\omega_1}\otimes V_{\omega_2}\otimes V_{\omega_3}\otimes V_{\omega_5}$ contains a subrepresentation of the form $V_{2\omega_1 + \omega_4+ 2\omega_5}$ Nov 28, 2022 at 18:06

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 $$\mathfrak{g}$$.