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Fix a finite group $G$, and look at all its irreducible representations/$\mathbb{C}$. It is said in Serre's book that "there cannot be any $\mathbb{C}$-linear relation between the matrix coefficients of these irreducible representations (of course we fix a set of bases first), because of the orthogonality formulas". I don't understand the quoted part. Can anyone explain that? I think I must be missing something obvious here.

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up vote 3 down vote accepted

Essentially because a set of pairwise orthgonal nonzero vectors must be linearly independent.

Here orthogonality is with respect to a complex (sesquilinear) positive-definite inner product, and we can regard each matrix entry as a vector of length $|G|$.

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Thanks! Crazy me, I have been thinking about characters and how to make the coefficients the same.. – Soarer Sep 22 '10 at 3:46

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