Suppose $\{e_1,\dots,e_n\}$ are the standard unit vectors in $\mathbb{R}^n$. Then the root system of type $B_n$ consists of $\pm e_i$, and $\pm(e_i\pm e_j)$ for $i\neq j$.
I know the Weyl group $W$ is $(\mathbb{Z}/(2))^n\rtimes S_n$. How explicitly does this act on the roots? For instance, if $((x_1,\dots,x_n),\sigma)\in W$, what does that do to a roots? Does it do something like $$ ((x_1,\dots,x_n),\sigma))(e_i)=(-1)^{x_{\sigma(i)}}e_{\sigma(i)} $$
where it changes the index of $e_i$, and then changes the sign depending on whether the coordinate $x_{\sigma(j)}$ is $0$ or $1$ in $(\mathbb{Z}/(2))^n$?