In some unpublished notes which are not publically available, a version of the following result is proved:
If $\phi_1$ and $\phi_2$ are complex, irreducible representations of compact Lie groups $G_1$ and $G_2$, then the exterior tensor product $\phi_1\boxtimes \phi_2$ is an irreducible representation of the direct product $G_1\times G_2$.
I am using this result in my thesis, but I don't want to write up a full proof because I would need some concepts that would need a proper explanation first, which would be out of the scope of my work.
I would like to have a published source to cite for this result, ideally a book. It looks like such a simple result that I am sure it must be proved in some representation theory book out there, but I haven't been able to find one.
Does anyone know of any book containing this (or a similar) statement?
Thank you