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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 $\sigma$?

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    @Mariano: how silly of me, of course, it's just the restriction of the action that comes with the representation :) Thank you. If you add it as an answer I'll accept it. @joriki: in conseque$nc$e, wh$a$t I s$a$id $a$bout th$a$t m$a$p is wro$n$g :P2011-02-09

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She is considering a complex representation V which comes with an action of the group elements on V.