Length and longest Weyl-group element

Let $\mathrm{\Phi }\subseteq E$ be a reduced crystallographic root system with positive system ${\mathrm{\Phi }}^{+}$ and simple roots $\mathrm{\Delta }$ (Positive systems and simple roots), and let $W=W(\mathrm{\Phi })$ be its Weyl group (Weyl group). For $w\in W$ define the inversion set $N(w)=\{\alpha \in {\mathrm{\Phi }}^{+}:w(\alpha )\in {\mathrm{\Phi }}^{-}\}$ and the length $\mathrm{\ell }(w)=\mathrm{\mid }N(w)\mathrm{\mid }$, the number of positive roots sent by $w$ to negative roots.

A longest element of $W$ is an element $w_0\in W$ with $\mathrm{\ell }(w_0)\ge \mathrm{\ell }(w)$ for all $w\in W$. The next proposition identifies $\mathrm{\ell }(w)$ with the minimum length of an expression of $w$ as a product of simple reflections $s_{{\alpha }_i}$, and proves that a longest element exists, is unique, and is characterized by $w_0({\mathrm{\Phi }}^{+})={\mathrm{\Phi }}^{-}$; its length is $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$. The length depends on the chosen positive system, hence on the chamber; replacing ${\mathrm{\Phi }}^{+}$ by $-{\mathrm{\Phi }}^{+}$ replaces the length function by $\mathrm{\ell }\circ (\cdot {)}^{-1}$ with respect to the new positive system.