Is there a criterion (or, more generally, an algorithm) to decide whether given non-negative integers $n_1, n_2$ and $m_1,...,m_k$ there is a group $G$ with irreducible reps $V_1, V_2$ over $\mathbb C$ of dimensions $n_1$ and $n_2$ whose tensor product $V_1\otimes V_2$ decomposes into $k$ irreducible components of dimensions $m_1,....,m_k$? (I am also interested in versions of this problem when $G$ is required to be finite or Lie.)

I imagine someone studied that in the 100+ years of history of representation theory.

algorithmlurking here, if one starts with the well-known classifications of simple Lie algebras and their irreducible finite dimensional representations over $\mathbb{C}$. Like Victor I wouldn't expect anything easy if the $n_i$ and $m_j$ can be arbitrary. $\endgroup$ – Jim Humphreys Dec 10 '16 at 16:03