§ Even and odd functions through representation theory
Consider the action of on the space of functions .
given by , and . How do we write this in terms of irreps?
- On the even functions, since for even, we have that, and [since ], or , hence the action of is that of the trivial representation on the subspace spanned by even functions.
Since the even and odd functions span the space of all functions, as we can write any function as the
sum of an even part and an odd part . So,
we have described the action of in terms of subspaces which span the space, so we've found the irrep decomposition.
- On the odd functions, since , we have that hence , hence where is the sign representation!