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Let V be an irreducible representation of a finite group G.How to show that up to scalars,there is a unique Hermitian inner product on V preserved by G. i know of how to get an inner product. but i have no idea on the uniqueness part. i think i have to use schur's lemma in some way

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up vote 3 down vote accepted

An inner product is the same as a map from $V \to \bar{V}$: $\langle -, - \rangle$ corresponds to $v \to \langle -, v \rangle$. $G$-invariant inner product corresponds to $G$-invariant maps $Hom_G(V, \bar{V})$. What can you say about this space by Schur's lemma?

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how does it correspond to "G-invariant maps" –  K.Ghosh Jan 20 '13 at 7:45
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@K.Ghosh, try to work out how $G$ acts on $Hom(V, \bar{V})$, and what it means for a map to be $G$-invariant. –  Sanchez Jan 20 '13 at 7:47
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