§ Even and odd functions through representation theory

Consider the action of $\mathbb Z/ 2\mathbb Z$ on the space of functions $\mathbb R \to \mathbb R$. given by $\phi(0)(f) = f$, and $phi(1)(f) = \lambda x. f(-x)$. How do we write this in terms of irreps?

• On the even functions, since $e(x) = e(-x)$ for $e$ even, we have that, $\phi(0)(e) = e$ and $\phi(1)(e) = e$ [since $e(-x) = e(x)$], or $\phi(x)(e) = id(e)$, hence the action of $\phi$ is that of the trivial representation on the subspace spanned by even functions.

• On the odd functions, since $o(-x) = -o(x)$, we have that $\phi(1)(o)(x) = o(-x) = -o(x) = sgn(o)(x)$ hence $\phi(1)(o) = -o$, hence $\phi(x)(o) = sgn(x)(o)$ where $sgn$is the sign representation!

Since the even and odd functions span the space of all functions, as we can write any function $f$ as the sum of an even part $e_f(x) \equiv [f(x) + f(-x)]/2$ and an odd part $o_f(x) \equiv [f(x) - f(-x)]/2$. So, we have described the action of $\phi$ in terms of subspaces which span the space, so we've found the irrep decomposition.