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I'm having some trouble with the concept of the direct sum and product of representations.

Say I have two representations [itex]\rho_1 , \rho_2[/itex] of a group [itex]G[/itex] on vector spaces [itex]V_1, V_2[/itex] respectively. Then I know their direct sum and their product are defined as

[itex] \rho_1 \oplus \rho_2 : G \to GL\left(V_1 \oplus V_2 \right)[/itex]

with [itex] \left[\rho_1 \oplus \rho_2 \right] (g) (v_1,v_2) = (\rho_1 (g) v_1 ,\rho_2 (g) v_2)[/itex]

and

[itex]\rho_1 \otimes \rho_2 : G to GL\left(V_1 \otimes V_2 \right) [/itex]

with [itex]\left[ \rho_1 \otimes \rho_2 \right] (g) v_1 \otimes v_2 =\rho_1 (g) v_1 \otimes \rho_2 (g) v_2 [/itex].

However, in calling these maps representations I have to show these are homomorphisms, which I can't seem to do. Can anyone help?

EDIT: Nevermind, I worked it out.

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# Direct sum and product representation

Can you offer guidance or do you also need help?

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