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Say I have two irreducible representations [itex]\rho[/itex] and [itex]\pi[/itex] of a group [itex]G[/itex] on vector spaces [itex]V[/itex] and [itex]W[/itex]. Then I construct a tensor product representation

[itex]\rho \otimes \pi : G\to \mathrm{GL}\left(V_1 \otimes V_2\right)[/itex]

by

[itex]\left[\rho \otimes \pi \right] (g) v\otimes w = \rho (g) v \otimes \pi (g) w [/itex].

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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# Reducibility tensor product representation

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