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Is there an example of the following situation : $F$ is a field, $G$ is a finite group, $\rho$ is an irreducible $F$-representation of $G$ and the character of this representation takes the zero value at every element of $G$ ? If I'm not wrong, $F$ cannot be algebraically closed (Robinson, A Course in the Theory of Groups, 8.1.9, p. 220) and must have a nonzero characteristic $p$ such that the degree of the representation is divisible by $p$ and such that $G$ is not a $p$-group (Robinson, exerc. 8.1.5, p. 222). But that doesn't solve the problem. Thanks in advance for the answers.

(Edit. Since I didn'get answers here, I asked the question on Mathoverflow :

https://mathoverflow.net/questions/252855/can-an-irreducible-representation-have-a-zero-character )

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  • $\begingroup$ Won't $\rho(1) = I$ the identity of $GL_n(F)$ ? $\endgroup$
    – reuns
    Oct 23, 2016 at 7:01
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    $\begingroup$ Yes, so the value of the character at 1 is the trace of the identity and this trace is zero if the degree of $\rho$ is divisible by $p$ (the characteristic of $F$). $\endgroup$
    – Panurge
    Oct 23, 2016 at 7:14
  • $\begingroup$ SImple remark: the $F$-algebra generated by $\rho(G)$ will also have trace identically zero, and will be a simple $F$-algebra. $\endgroup$
    – YCor
    Oct 23, 2016 at 7:42
  • $\begingroup$ @YCor Why is it clear that the trace is zero? If that is true, we are done, because I can reduce to the case that $F$ is a finite field. All simple algebras over a finite field are separable field extensions (Wedderburn's little theorem + finite fields are perfect) and the trace of a separable field extension is nonzero. $\endgroup$
    – David E Speyer
    Oct 23, 2016 at 15:02
  • $\begingroup$ @DavidSpeyer because the $F$-subspace generated by a subgroup of $GL_n(F)$ is stable under multiplication, and hence equals the $F$-subalgebra it generates. $\endgroup$
    – YCor
    Oct 23, 2016 at 15:40

1 Answer 1

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CW answer to take this off the unanswered list. The question was answered by the OP on MO. The answer is "no"; see I.M. Isaacs, Character Theory of Finite Groups, coroll 9.22, Dover, p. 155.

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