Suppose $G$ is a finite group and $\mathbb{k}$ an algebraically closed field of characteristic zero. Denote the irreducible representations of $G$ over $\mathbb{k}$ by $V_1,\ldots, V_t$. Is it necessarily true that $\sum\limits_{i=1}^t\chi_{V_i}(g)^2\neq 0$ for $g\in G$?
On a sum taken over all irreducible characters: is $\sum\limits_{i=1}^t\chi_{V_i}(g)^2$ nonzero for all $g\in G$?
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finite-groups
representation-theory
characters
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1Try the group of order 3. – 2012-10-11
2 Answers
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It doesn't seem to be true for a generator of $C_3$ (and $\mathbb k=\mathbb C$).
Then $\chi_{V_i}(g)$ are the three cube roots of unity, and so are their squares.
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1Wow, seems obvious now. Thanks – 2012-10-12
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You may be thinking of $\displaystyle\sum_{i=1}^t |\chi_{V_i}(g)|^2$, which is equal to $|C_G(g)|$ and thus always positive.
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1You're right, thank you – 2012-10-12