# Coxeter element

(Redirected from Coxeter number)
Not to be confused with Longest element of a Coxeter group.

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group, hence also of a root system or its Weyl group. It is named after H.S.M. Coxeter.[1]

## 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 mirrors in the Coxeter group is half the number of roots.
• The Coxeter number is the order of a Coxeter element; note that conjugate elements have the same order.
• If the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi
• The dimension of the corresponding Lie algebra is n(h + 1), where n is the rank and h is the Coxeter number.
• The Coxeter number is the highest degree of a fundamental invariant of the Weyl group acting on polynomials.
• The Coxeter number is given by the following table:
Coxeter group Coxeter number h Dual Coxeter number Degrees of fundamental invariants
An ... n + 1 n + 1 2, 3, 4, ..., n + 1
Bn ... 2n 2n − 1 2, 4, 6, ..., 2n
Cn n + 1
Dn ... 2n − 2 2n − 2 n; 2, 4, 6, ..., 2n − 2
E6 12 12 2, 5, 6, 8, 9, 12
E7 18 18 2, 6, 8, 10, 12, 14, 18
E8 30 30 2, 8, 12, 14, 18, 20, 24, 30
F4 12 9 2, 6, 8, 12
G2 = I2(6) 6 4 2, 6
H3 10 2, 6, 10
H4 30 2, 12, 20, 30
I2(p) p 2, p

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 ei(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, ζh = ei/h, which is important in the Coxeter plane, below.

## Coxeter elements

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

The dihedral group Dihm is generated by two reflections that form an angle of $2\pi/2m$, and thus their product is a rotation by $2\pi/m$.

## Coxeter plane

Projection of E8 root system onto Coxeter plane, showing 30-fold symmetry.

For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h. This is called the Coxeter plane and is the plane on which P has eigenvalues ei/h and e−2πi/h = ei(h−1)/h.[3] This plane was first systematically studied in (Coxeter 1948),[4] and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.[4]

The Coxeter plane is often used to draw diagrams of higher-dimensional polytopes and root systems – the vertices and edges of the polytope, or roots (and some edges connecting these) are orthogonally projected onto the Coxeter plane, yielding a Petrie polygon with h-fold rotational symmetry.[5] For root systems, no root maps to zero, corresponding to the Coxeter element not fixing any root or rather axis (not having eigenvalue 1 or −1), so the projections of orbits under w form h-fold circular arrangements[5] and there is an empty center, as in the E8 diagram at above right. For polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids.