Given a fundamental representation $V(\nu\_k)$ of a semisimple Lie algebra $\frak{g}$, and a general irreducible finite-dimensional representation $V$, is it ever possible that the tensor product $V \otimes V(\nu\_k)$ can contain a copy of $V$?



Tensoring $\frak{g}$-modules by fundamental representations
Yes. First of all, if we take a fixed irrep $ V(\mu) $, then for big enough $ \lambda$, we know that $ V(\lambda + \nu) $ will appear in $ V(\lambda) \otimes V(\mu) $ with multiplicity equal to the weight multiplicity of $ \nu $ in $V(\mu) $.


So your question is equivalent to: is $ 0 $ a weight of a fundamental representation? For $ \mathfrak{sl}\_n $, this is not possible, but for other simple Lie algebras, this is possible. (Is it possible for every simple Lie algebra?)


