Reducibility tensor product representation

[Yoran91]

Hello everyone,

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.

 

[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.