Let $B_n$ be the group of signed permutations, which is a Coxeter group acting on $\mathbb{R}^n$ with Coxeter generators $\sigma_i=(i\; i+1)\in S_n$ and the change of sign $\tau(x_1,x_2,\dots,x_n)=(-x_1,x_2,\dots,x_n)$.
So the elements can be represented by the action on the vector $(1,2,\dots,n)$ as words $w$ in the signed alphabet $\{\pm 1,\dots,\pm n\}$ where the $|w_i|$ form a permutation. Moreover, $$ \operatorname{inv}(w)=|\{i
How does that characterization follow from this definition? Thank you.