# Reducibility tensor product representation

1. May 26, 2013

### Yoran91

Hello everyone,

Say I have two irreducible representations $\rho$ and $\pi$ of a group $G$ on vector spaces $V$ and $W$. Then I construct a tensor product representation
$\rho \otimes \pi : G\to \mathrm{GL}\left(V_1 \otimes V_2\right)$
by
$\left[\rho \otimes \pi \right] (g) v\otimes w = \rho (g) v \otimes \pi (g) w$.

I now wish to know whether or not this representation is reducible or irreducible. If it cannot be determined, then I wish to know what further conditions imply reducibility or irreducibility. However, I have not been able to find an answer to this anywhere. Can anyone provide some insight?

Thanks for any help.

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