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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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Bruno, she is considering a complex representation $V$ which comes with an action of the group elements on $V$. – Mariano Suárez-Alvarez Feb 9 '11 at 19:46
Yes, that map $\langle\sigma\rangle \times V \rightarrow V$ is the action. You can find more here: – joriki Feb 9 '11 at 20:09
@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 consequence, what I said about that map is wrong :P – Bruno Stonek Feb 9 '11 at 21:12
up vote 3 down vote accepted

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

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