Coxeter element
In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order.
There are many different ways to define the Coxeter number h of an irreducible root system.
A Coxeter element is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order.
The Coxeter number is the number of roots divided by the rank. The number of reflections in the Coxeter group is half the number of roots.
The Coxeter number is the order of any Coxeter element;.
If the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi
The Coxeter number is the dimension of the corresponding Lie algebra is n(h + 1), where n is the rank and h is the Coxeter number.