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Math Help - (Ir)reducibility of tensor product representation

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    (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!
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    MHF Contributor Drexel28's Avatar
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    Re: (Ir)reducibility of tensor product representation

    Quote Originally Posted by Lockdown View Post
    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
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