Can a character of a finite group be identically zero, so is $\chi=0\times\chi_1+\cdots+0\times\chi_n$ possible?

Question

I was wondering if the following case is possible?

Suppose we have a finite group $G$ and suppose I know a complete list of irreducible characters of $G$, say $\{\chi_1,\cdots\chi_n\}.$ Also I am aware that every character of $G$ can be written as $\sum a_i\chi_i$ where $a_i$ are all non negative integers. I was wondering can $a_i's$ be all $0$?

So I have got a few questions:

• Is this possible?

• If it is, what kind of situation does it correspond to? Do we have that the corresponding $\mathbb{C}G$- module is a trivial module?

I am really not sure how to interpret this situation, could someone please enlighten me?

Thank you so much in advance!

Answer 1

Notice that $\chi(1)=\operatorname{tr}(id_V)$, therefore such $\chi$ can only correspond to a $0$-dimensional representation sending the whole $G$ to the identity of $\Bbb F^0$. Not sure if it's an interesting case to inspect.