a function as a character 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.

I think here is the solution : Looking the second question, it suggests us to prove that $\chi_Y$ is a representation, i.e., to prove that $\chi_Y$ is a group homomorphism.
Let $v_1,v_2\in(\mathbb{Z}/2\mathbb{Z})^m$ so
$\alpha(v_1+v_2)=\alpha(v_1)+\alpha(v_2)-\alpha(v_1)\cap\alpha(v_2)$, and so
$|\alpha(v_1+v_2)\cap Y| = |\alpha(v_1)\cap Y|+|\alpha(v_2)\cap Y| -2|(\alpha(v_1)\cap\alpha(v_2))\cap Y|$, so the results follows.
For (2), Since $(\mathbb{Z}/2\mathbb{Z})^m$ is abelian, every irreducible representation is linear, let $\phi$ be a such one, we need to prove there is a set $Y_\phi$ associated with $\phi$. Since every irreducible representation of $(\mathbb{Z}/2\mathbb{Z})^m$ comes from the irreducible representations of $\mathbb{Z}/2\mathbb{Z}$, which are $1$ and $\rho$, where $\rho(0)=1$ and $\rho(1)=-1$, i.e., $\phi$ is a product $f_1\cdots f_m$ where $f_i$ is either $1$ or $\rho$. Then $Y_\phi=\{i : f_i=\rho\}$.
