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Math Help - Irreducible representation

  1. #1
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    Irreducible representation

    Let G be a group and let z be in the center of G.
    Why is \rho(z) for every irreducible representation \rho of G a scalair?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bram kierkels View Post
    Let G be a group and let z be in the center of G.
    Why is \rho(z) for every irreducible representation \rho of G a scalair?
    What do you define as rho?
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    What do you define as rho?
    rho is just an arbitrary representation, thus \rho:G\rightarrow GL(V) is a homomorphism on a (complex) vector space V, where GL(V) is the general lineair group.
    I think it is easier then it looks:
    Since \rho(z):V\rightarrow V, and \rho(z)\rho(x)=\rho(zx)=\rho(xz)=\rho(x)\rho(z)., it follows directly from this corollary of Schur's Lemma:
    PlanetMath: Schur's lemma
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