Can an irreducible representation have a zero character? I asked the following question on Stackexchange,
https://math.stackexchange.com/questions/1978407/can-an-irreducible-representation-have-a-zero-character
but it got no answer, so I ask it here.
Is there an example of the following situation : $F$ is a field, $G$ is a finite group, $\rho$ is an irreducible $F$-representation of $G$ (with finite degree) 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.
 A: Well, it seems that the character of an irreducible representation of a finite group over whatever field is always nonzero. I find this statement in I.M. Isaacs, Character Theory of Finite Groups, coroll 9.22, Dover, p. 155.
A: Representations of semigroups whose characters are identically zero  have been classified in 
A. Freedman, R. N. Gupta, and R. M. Guralnick, Shirshov’s theorem and representations of semigroups, Pacific J. Math. (1997), no. 181, 159--176.
Their result is the following, see Theorem 2.5 in the aforementioned paper:

Theorem. Let $\rho \colon S \to \mathrm{End}(V )$ be a finite irreducible representation
  of the semigroup $S$ and  let $A$ be the $F$-algebra generated by $\rho(S) \subseteq  \mathrm{End}(V )$.
Then the character $\chi$ of $\rho$ is identically $0$ if and only if $A \simeq M_r(D)$, where
  $D$ is division ring such that either $Z = Z(D)$ is not separable over $F$ or
  $\mathrm{dim}_Z(D)$ is a multiple of the characteristic. In particular, if $F$ is perfect, the character in nonzero.

${}$
Edit. As explained by B. Steinberg in the comments below, none of the conditions in the previous Theorem can occur when $S$ is a finite semigroup. In particular, when $G$ is a finite group over any field $F$, the character of any irreducible $F$-representation of $G$ is non-zero. This agrees with the result in Isaac's book quoted in Panurge'a answer.
