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Math Help - [SOLVED] Irreducible Representations

  1. #1
    Lord of certain Rings
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    [SOLVED] Irreducible Representations

    Let V be a finite dimensional representation of a finite group G. Consider Hom(V,V) = Set of all Linear Transformations from V to V. Its clearly a vector space (under the natural operations).

    Prove that : Hom(V,V) is an irreducible representation of G \Leftrightarrow dim V = 1

    The problem for me is that I cant understand how the group GL(Hom(V,V)). It looks extremely complicated.

    Trying to prove: \Leftarrow
    I assume ( \phi,V) is the representation over the field F, dim V = 1. Then Hom(V,V) \cong F.
    F is a one-dimensional vector space over itself and thus Hom(V,V) is one dimensional and hence irreducible.

    To prove: \Rightarrow
    I am struggling here. So any ideas?

    Thank you,
    Srikanth
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  2. #2
    Lord of certain Rings
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    Hello ppl,

    Well... I solved it anyway. But I am still open to comments >_<

    Proof of \Rightarrow

    \psi: G \to \text{GL(Hom(V,V))}, \psi_g(T) = \phi_g \circ T \circ \phi_{g}^{-1}

    Now consider the subspace of scalar matrices W. We can easily see that \psi_g(\lambda \mathbb{I}_V) = \phi_g \circ \lambda \mathbb{I}_V \circ \phi_{g}^{-1} = \lambda \mathbb{I}_V \in W. Thus W is non zero G-invariant subspace of Hom(V,V). But data says Hom(V,V) is irreducible and thus W = Hom(V,V). But W is clearly 1 dimensional ( \mathbb{I}_V is a basis)
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