Direct sum and product representation

[Yoran91]

Hi everyone,

I'm having some trouble with the concept of the direct sum and product of representations.
Say I have two representations \rho_1 , \rho_2 of a group G on vector spaces V_1, V_2 respectively. Then I know their direct sum and their product are defined as

\rho_1 \oplus \rho_2 : G \to GL\left(V_1 \oplus V_2 \right)
with \left[\rho_1 \oplus \rho_2 \right] (g) (v_1,v_2) = (\rho_1 (g) v_1 ,\rho_2 (g) v_2)

and

\rho_1 \otimes \rho_2 : G \to GL\left(V_1 \otimes V_2 \right)
with \left[ \rho_1 \otimes \rho_2 \right] (g) v_1 \otimes v_2 =\rho_1 (g) v_1 \otimes \rho_2 (g) v_2.

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.

 

[fresh_42]

It's a straightforward calculation. The trick with representations is to keep track of where you are: $G, V, GL(V)$.