# Weil's proof of a theorem on finite irreducible representations of products of compact groups

Theorem Let $G$ and $H$ be compact groups. Let $ρ$ be a finite dimensional irreducible continuous representation of $G×H$ over the field of complex numbers. Then $ρ$ is a tensor product of irreducible representations of $G$ and $H$.

In his book L'integration dans les groupes topologiques, Weil proved this theorem under more general conditions. His proof was short and elementary. He used no functional analysis. On the other hand, Pontryagin proved the same theorem using the Peter-Weyl theorem in his famous book.

I was puzzled. Is Weil's proof correct?

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Since $\rho$ is finite-dimensional, once restricted to $G = G\times 1,$ we can find an irreducible $G$-subrep'n $\psi$. Consider $Hom_G(\psi,\rho)$; this is a finite-dimensional $H = 1\times H$-representation. Let $\chi$ be an irreducible $H$-subrep'n. Then evaluation gives a non-zero map $\psi \otimes \chi \hookrightarrow \rho,$ which must be an isomorphism, since the source and target are both irreducible.
@Jason: Dear Jason, There is a typo (and thanks for drawing my attention to it). I meant to write "we can find an irreducible $G$-subrep'n $\psi$''. Correction to ensue! Regards, – Matt E Apr 24 '12 at 2:23