About expressing $w\rho-\rho$ as a sum of roots

Question

Let $\Phi$ be a root system, $\Phi^+$ be the positive system, $\rho$ be the half sum of positive roots, and $W$ be the Weyl group of $\Phi$.

I remember that there is a way to express $w\rho-\rho$ as a sum of roots for each $w\in W$. However, I forget the detail. Does anyone know that?

Answer 1

Let $N(w)$ be the inversion set of the element $w \in W$, i.e. the set

$\{\alpha \in \Phi^+ s.t. w(\alpha) \in \Phi^-\}$

Then it can be easily proved by induction on the lenght of $w$ that

$\rho - w\rho = \sum_{ \alpha \in N(w) }\alpha$