Consider a unitary irrep of a compact Lie Group $G$ onto a Hilbert space $\mathcal H$, $\pi\colon G\to\mathcal U(\mathcal H)$. Now assume that $\mathcal H$ can be decomposed into a tensor product, $\mathcal H=V\otimes W$. Is it possible that $V$ and $W$ are also representaions of $G$?
In other words, can a unitary irrep be decomposed into a tensor product of (reudcible or not) representations? Maybe exluding one-dimensional tensor factors?
Edit: Note that I don't want to assume more than compactness of the Lie group $G$, in particular no direct product struture...
As the commentors on to the question have rightly pointed out, the quesiton as it is posed is indeed answered by irreducibility of a tensor product of two irreducible representations. Thanks, @user10354138 and @QiaochuYuan.