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A Irreducibility of (anti)self-dual reps

  1. May 12, 2016 #1
    Imagine we are talking about the group SO(4). The second rank antisymmetric representation is reducible into self-dual and antiself-dual representations. I think a good way to visualise this is by noticing that the projection of [itex] \Lambda^{2}V [/itex] into self and antiself dual subspaces commutes with the action of SO(4).

    However, how can I show that those subspaces are themselves irreducible?

    Thanks!
     
  2. jcsd
  3. May 12, 2016 #2

    fresh_42

    Staff: Mentor

    It would be easier to argue if you supply a more detailed framework. In general a representation is irreducible if there are no proper submodules. So you could take a basis vector and show that its orbit is the whole module.
     
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