Coxeter length ($\ell (w)$)

The Coxeter length $\ell (w)$ is the smallest number of simple reflections whose product is $w$. Its parity gives the sign of a Weyl-group element.

This is a particular length function, namely word length for the simple reflections.

If $\lambda$ and $w\lambda$ both lie in the closed dominant chamber, then $w$ is a product of simple reflections whose walls contain $\lambda$. In particular, $w\lambda =\lambda$.

For a chosen positive system of a root system $\Pi$, the inversion set isFor a finite Weyl group, $|N(w)|=\ell (w)$.

Its cardinality is the Coxeter length; this root-sign definition should not be silently identified with an unspecified permutation inversion convention.

In the geometric representation of a Coxeter system,andReplacing “positive” by “negative” reverses either length inequality.