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1)Let [tex]\pi_i[/tex] be representations of a group G on vector spaces V

_{i}, i = 1, 2. Give a formula for the tensor product representation [tex]\pi_1 \otimes \pi_2[/tex] on [tex]V_1 \otimes V_2[/tex]

**Answer:**[tex]\pi_1 V_1 \otimes \pi_2 V_2[/tex]

2)Check that it obeys the representation property.

**Answer:**A representation is a group homomorphism, ie it satisfies:

[tex]\pi(g.h)= \pi(g) . \pi(h)[/tex]

Now,

[tex]

[\pi_1 V_1 \otimes \pi_2 V_2](g.h)

=\pi_1 V_1 (gh) \otimes \pi_2 V_2 (gh)

[/tex]

I am a little stuck here: we know that [tex]\pi_i[/tex] is a representation, can we also say that [tex]\pi_i V_i[/tex] is also a representation? If it is, we can use the homomorphism property and show that

[tex] \pi_1 V_1 (gh) \otimes \pi_2 V_2 (gh)=\pi_1 V_1 (g)\pi_1 V_1 (h) \otimes \pi_2 V_2 (g) \pi_2 V_2 (h)=[\pi_1 V_1 \otimes \pi_2 V_2](g)[\pi_1 V_1 \otimes \pi_2 V_2](h)[/tex]

which I think the question is trying to get at.