Let $\mathfrak{g}$ be a semi-simple finite dimensional Lie algebra. Denote by $L(\lambda)$ an irreducible finite-dimensional $\mathfrak{g}$-module of highest weight $\lambda$. (I.e. $\lambda$ is integral dominant weight) Suppose that the multiplicity of $L(\nu)$ in the tensor product $L(\lambda)\otimes L(\mu)$ is greater than $0$.

Does this imply that either $\lambda \geq \nu$ or $\lambda \leq \nu$? Where the comparison is the standard partial ordering on the set of weights: $\lambda \geq \nu$ iff their difference $\lambda-\nu$ is equal to the sum of positive roots with nonnegative coefficients.

`$\nu=0$`

: take the natural module`$V$`

for`$\mathfrak{sl}_n$`

, with highest weight`$\varpi_1$`

, so the trivial module occurs as a summand of`$\mathrm{End}(V) \cong V^* \otimes V$`

. All you need here is a highest weight not in the root lattice, so 0 isn't a subweight. $\endgroup$ – Jim Humphreys Jun 17 '13 at 18:49