Let $\frak{g}$ be a complex semisimple Lie algebra with root lattice $Q$ and positive weight space $P^+$. Let $\lambda, \mu \in Q \cap P^+$, with corresponding respective fin-dim irreducible representations $V\_{\lambda}$ and $V\_{\mu}$. Form the tensor product $V\_{\mu} \otimes V\_{\lambda}$ and then decompose it into irreducibles
$$
V\_{\mu} \otimes V\_{\lambda} \simeq V\_{\gamma\_1} \oplus \cdots \oplus V\_{\gamma\_m}.
$$
Will the dominant weights $\gamma\_i$ still be contained in $Q \cap P^+$ or will they in general only be contained in $P^+$? If they are only contained in $P^+$, then what is a simple example that illustrates this?


Finally, if in general there exist such $\gamma$'s that live outside $Q \cap P^+$, can we categorize/classify the subset of all such dominant weights?



Tensoring irreducible representations corresponding to root lattice elements
Just to summarize what was mentioned in the comments ([1](https://mathoverflow.net/questions/440348/tensoring-irreducible-representations-corresponding-to-positive-root-lattice-ele#comment1136048_440348) [2](https://mathoverflow.net/questions/440348/tensoring-irreducible-representations-corresponding-to-positive-root-lattice-ele#comment1136051_440348)) and to have this question marked as answered, what follows is the explanation of why the answer is *yes*, the irreps that appear in the decomposition of $V\_{\mu}\otimes V\_{\lambda}$ will be indexed by root lattice elements.


First notice that every weight that appears in both $V\_{\mu}$ and $V\_{\lambda}$ must also belong to the root lattice (because we can get to every weight from the highest weight by going down along simple roots: this is a well-known fact about the structure of irreps for semisimple Lie algebras). Then note that $V\_{\mu}\otimes V\_{\lambda}$ is spanned by $u\otimes v$ where $u$ is a weight basis element of $V\_{\mu}$ and $v$ a weight basis element of $V\_{\lambda}$. But the weight of $u\otimes v$ is the sum of the weights of $u$ and of $v$. So all the $u\otimes v$ have weights that belongs to the root lattice. In particular, all highest weight vectors do too, hence all irreps that appear are indexed by root lattice elements.


