# Representation of $S_3$ on $\mathbb{C}^6$.

I am asked to decompose $S_3$ into its irreducible unitary representations, on $\mathbb{C}^6$.

I wonder how does this differ from analysis the given in the next paper?: http://users.jyu.fi/~laurikah/REP/REPtext2010_2.pdf (pages 50-51).

Thanks.

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What you are asking for is a decomposition of the regular representation of $S_3$ (I suppose). The answer is, in general, that every possible irreducible representation occurs with the multiplicity equal to its dimension. So in this case the trivial and the sign representation both once, the two-dimensional irreducible representation (action on a triangle) twice.