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Completeness and orthogonality of unitary irreducible representations

  1. Feb 15, 2016 #1
    I'm reading Lie Algebras and Particle Physics by Howard Georgi. He is trying to prove (section 1.12) that the matrix elements of the unitary irreducible representations (irreps) form a vector space of dimension N where N is the order of the group. For example for the matrix of the kth unitary irrep, consider element ij, it can be described as an N tuple of complex numbers and is therefore a vector in an N dim space. He shows that the collection of all such matrix elements form an orthonormal set of vectors.

    Now he has to show that these vectors span an N dimensional space. He does this by first raising the point that the space of functions f:G--->C is N dimensional since in the regular representation we can write every function on the group elements as a functional, or a covector. I don't see however how this space is spanned by the vectors from the orthonormal proof.

    Can anyone please help me out, and also I need some confirmation that what I think I understand, I actually got right.

  2. jcsd
  3. Feb 20, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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