I meet difficulty in Problem 4.5 in the book "Representation theory of finite group, an introductory approach" of Benjamin Steinberg :
For $v=(c_1,\cdots,c_m)\in(\mathbb{Z}/2\mathbb{Z})^m$, let $\alpha(v)=\{i : c_i=[1]\}$. For each $Y\subseteq\{1,\cdots,m\}$, define a function $\chi_Y : (\mathbb{Z}/2\mathbb{Z})^m \to \mathbb{C} $ by $\chi_Y(v)=(-1)^{|\alpha(v)\cap Y|}$.
- Prove that $\chi_Y$ is a character.
- Show that every irreducible character of $(\mathbb{Z}/2\mathbb{Z})^m$ is of the form $\chi_Y$ for some subset $Y$ of $\{1,\cdots,m\}$.
Thanks in advance.