I am currently taking a representation theory class (from a physicist), and I am very confused about the Dirac groups' irreducible representations.

First of all, all the Dirac matrices in the representation have trace = 0, so it does not even seem to include a unit matrix. When we talked about representations in class before, we always had a unit matrix in a representation, what happened?

Also, the lecture went into distinguishing the case for 2n-dimension and 2n-1-dimension. While I understand why there is one more conjugacy classes in the odd dimension (thus even dimension having one more irreducible rep than odd dimension), I can't fully appreciate all the difference in the irreducible representations in the cases of even and odd dimensions; in particular, I was asked in a homework to show that if a Dirac matrices {$\gamma^\mu$} form an irreducible rep, then show that {$-\gamma^\mu$} is equivalent irreducible rep in the case of even dimension, and inequivalent irreducible rep in the case of odd dimension. But then again, if I think about the character table to see whether a representation is equivalent or inequivalent to another representation, I feel like matrices in {$\gamma^\mu$} and {-$\gamma^\mu$} will never have same trace, thus they can never be equivalent (unless they are all 0, which is the case, I believe. Then again, how could they be different representations then?).

I would appreciate any good reading materials / answers to my questions!

This post imported from StackExchange Physics at 2016-04-05 08:55 (UTC), posted by SE-user Quantization