Completeness and orthogonality of unitary irreducible representations

[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

 

[jvicens]

Hello there,

Thank you for sharing your thoughts on this topic. As a scientist who is familiar with Lie algebras and particle physics, I would like to provide some clarification and confirmation on your understanding.

Firstly, I agree with you that the vector space formed by the matrix elements of the unitary irreducible representations is indeed of dimension N, where N is the order of the group. This is because each matrix element can be described as an N-tuple of complex numbers, which can be seen as a vector in an N-dimensional space. And as you mentioned, Georgi shows that these matrix elements form an orthonormal set of vectors.

To address your question about how these vectors span an N-dimensional space, let me explain it in more detail. The regular representation of a group G is the set of all functions f: G -> C, where C is the set of complex numbers. In other words, it is the space of all possible functions on the group elements. Now, for each function f in this space, we can define a corresponding functional, or covector, by taking the inner product with each of the matrix elements from the unitary irreducible representations. This results in an N-dimensional space, as there are N matrix elements in each unitary irrep.

So, in essence, the orthonormal set of matrix elements spans this N-dimensional space of functions on the group elements, which is the regular representation. This can be seen as a consequence of the fact that the matrix elements form a complete and orthonormal set of vectors.

I hope this helps clarify your understanding. If you have any further questions, please feel free to ask. Keep up the good work in your studies!