Coxeter plane

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.

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 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^{2πi(m − 1)/h} as m runs through the degrees of the fundamental invariants. Since this starts with m = 2, these include the primitive hth root of unity, ζ_h = e^2πi/h, which is important in the Coxeter plane, below.

There are relations between group order, g, and the Coxeter number, h:

An example, [3,3,5] has h=30, so 64*30/g = 12 - 3 - 6 - 5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2= 960*15 = 14400.

Coxeter elements of ${\displaystyle A_{n-1}\cong S_{n}}$, considered as the symmetric group on n elements, are n-cycles: for simple reflections the adjacent transpositions ${\displaystyle (1,2),(2,3),\dots }$, a Coxeter element is the n-cycle ${\displaystyle (1,2,3,\dots ,n)}$.

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