In these Karen Smith's notes on representation of finite groups, on page 50 the irreducible representations of $S_3$ are found.
If $\sigma=(1 2 3)$ and $V$ is a complex representation of $S_3$, I don't understand what "the action of $\sigma$ on $V$" means. Is it this:
$\langle\sigma\rangle \times V \to V$
$\sigma \cdot (v_1,\dots,v_n)=(v_{\sigma(1)},v_{\sigma(2)},v_{\sigma(3)},v_4,\dots,v_n)$
where $\langle\sigma\rangle$ means the subgroup of $S_3$ generated by $\sigma$?