Reducibility tensor product representation

[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.

 

[fresh_42]

I cannot imagine that (in general) irreducibility will be conserved. The new components could lie somehow diagonal in $V\otimes W$. For linear groups there is e.g. a theorem which says: If $\lambda$ is a dominant weight according to a maximal torus of $G$, that is all coefficients of $\lambda$ are non-negative, then there is an irreducible $G$ module of highest weight $\lambda$.

See: James E. Humphreys, Linear Algebraic Groups.

Your question is in its generality too broad to be answered as it depends on unknowns as which groups, or which fields.