I just started to read the book 'Representation theory of finite groups' by B. Steinberg and I'm trying to solve Exercise 4.5., which discusses the characters of $(\mathbb{Z}/2\mathbb{Z})^m$.
For a vector $v=(v_1,\dots,v_m)\in(\mathbb{Z}/2\mathbb{Z})^m$, let $\alpha(v)$ denote the set of indices $i\in\{1,\dots,m\}$ with $v_i=1$. For every subset $Y$ of $\{1,\dots,m\}$, we define the function $\chi_Y\colon(\mathbb{Z}/2\mathbb{Z})^m\rightarrow\mathbb{C}$ by
$$ \chi_Y(v)=(-1)^{|\alpha(v)\cap Y|}.$$
I want to show that this function is a character but I don't really know how to do this. I know that a character of a representation is defined as the trace of this representation. So, do I have to construct a representation whose character is $\chi_Y$? Or, is there a different approach?
This character is already the representation, since it is the character of the 1-dimensional representation, $\rho_Y : (\Bbb{Z}/2\Bbb{Z})^m \to \Bbb{C}^\times=\operatorname{GL}_1(\Bbb{C})$, $\rho_Y(v) = \chi_Y(v)$. You just need to show that $\rho_Y$ is a valid representation (i.e., that it is a group homomorphism).