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Let $U$ be the trivial representation of $S_{4}$ and $V$ be the standard representation (of $S_{4}$). Why is it that $\mathbb{C}^{4} = U \oplus V$? I see that this is true by character theory, but is there a way to show it without character theory?

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You probably should define your terms more precisely. I presume you mean the natural permutation module when you mentioned $\mathbb{C}^4.$ – Geoff Robinson Feb 18 '12 at 17:27
up vote 3 down vote accepted

One way to see this, assuming you do mean the natural permutation representation by $\mathbb{C}^4$, is that the permutation matrices are unitary matrices. There is an obvious $1$-dimenional invariant subspace $U$, conisting of (say column) vectors with al coordinats equal. The action of $S_4$ on this space is trivial. Since permutation matrices are unitary, the orthogonal complement $U^{\perp}$ of $U$ with respect to the usual inner product is a ($3$-dimensional) invariant subspace, and we have $\mathbb{C}^4 = U \oplus U^{\perp}.$

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