Coxeter element

In mathematics, a Coxeter element is an element of an irreducible Coxeter group which 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. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups.^[1]

Not to be confused with Longest element of a Coxeter group.

The Coxeter number is the order of any Coxeter element;.

The Coxeter number is ${\tfrac {2m}{n}},$ where $n$ is the rank, and $m$ is the number of reflections. In the crystallographic case, $m$ is half the number of ; and $2 m + n$ is the dimension of the corresponding semisimple Lie algebra.

roots

If the highest root is $\sum m_{i}\alpha _{i}$ for simple roots $α i$ , then the Coxeter number is $1+\sum m_{i}.$

The Coxeter number is the highest degree of a fundamental invariant of the Coxeter group acting on polynomials.

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.

The Coxeter number for each Dynkin type is given in the following table:

The invariants of the Coxeter group acting on polynomials form a polynomial algebra whose generators are the fundamental invariants; their degrees are given in the table above. Notice that if $m$ is a degree of a fundamental invariant then so is $h + 2 - m$ .

The eigenvalues of a Coxeter element are the numbers $e^{2\pi i{\frac {m-1}{h}}}$ as $m$ runs through the degrees of the fundamental invariants. Since this starts with $m = 2$ , these include the primitive $h$ th root of unity, $\zeta _{h}=e^{2\pi i{\frac {1}{h}}},$ which is important in the Coxeter plane, below.

The dual Coxeter number is 1 plus the sum of the coefficients of simple roots in the highest short root of the dual root system.