# Completeness and orthogonality of unitary irreducible representations

1. Feb 15, 2016

### hideelo

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.

Thanks

2. Feb 20, 2016