# Thread: (Ir)reducibility of tensor product representation

1. ## (Ir)reducibility of tensor product representation

Hi everyone,

Say I have two irreducible representations $\pi_1$ and $\pi_2$ of some group $G$ on vector spaces $V_1$ and $V_2$ and I form a tensor product representation
$\pi_1 \otimes \pi_2 : G \to \mathrm{GL}\left(V_1\otimes V_2\right)$
with $\left[\pi_1 \otimes \pi_2 \right]\left(g\right) v_1 \otimes v_2 = \pi_1 \left(g\right)v_1 \otimes \pi_2 \left(g\right) v_2$.

Under what conditions is this tensor product representations reducible or irreducible?
Thanks for any help on this!

2. ## Re: (Ir)reducibility of tensor product representation

Originally Posted by Lockdown
Hi everyone,

Say I have two irreducible representations $\pi_1$ and $\pi_2$ of some group $G$ on vector spaces $V_1$ and $V_2$ and I form a tensor product representation
$\pi_1 \otimes \pi_2 : G \to \mathrm{GL}\left(V_1\otimes V_2\right)$
with $\left[\pi_1 \otimes \pi_2 \right]\left(g\right) v_1 \otimes v_2 = \pi_1 \left(g\right)v_1 \otimes \pi_2 \left(g\right) v_2$.

Under what conditions is this tensor product representations reducible or irreducible?
Thanks for any help on this!
Hello,

You haven't specified what field you are taking this with respect to? If it's $\mathbb{C}$ try just considering the fact that irreducible representations are those with unit length characters.

Best,
Alex