Let $G$ be a finite abelian group, and let $G\longrightarrow \hat{G}$, $g\mapsto \chi_g$ be a group isomorphism from $G$ to its dual (i.e. the abelian group of group morphisms from $G$ to $\textbf{C}^{\times}$). We define two endormorphisms of $V=\textbf{C}[G]$ as a $\textbf{C}$-vector space by their action on $g\in G$: $$\rho_S(g)=\frac{1}{\sqrt{|G|}}\sum_{h\in G}{\overline{\chi_h(g)}h},$$ $$\rho_{T^2}(g)=\chi_g(g)g.$$ I have several questions:
If $\Gamma$ is the matrix group generated by $S=\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$ and $T^2=\left(\begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix}\right)$, does $\rho:\Gamma\longrightarrow \operatorname{GL}(V)$ defines a representation?
Is this the so-called Weil representation attached to $G$?
What do we know about the reducibility of $\rho$?
Many thanks!