Coxeter element

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 order of any Coxeter element;.
• The Coxeter number is 2m/n, where n is the rank, and m is the number of reflections. In the crystallographic case, m is half the number of roots; and 2m+n is the dimension of the corresponding semisimple Lie algebra.
• If the highest root is Σm_iα_i for simple roots α_i, then the Coxeter number is 1 + Σm_i.
• The Coxeter number is the highest degree of a fundamental invariant of the Coxeter group acting on polynomials.

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

Coxeter group		Coxeter diagram	Dynkin diagram	Reflections m=nh/2[2]	Coxeter number h	Dual Coxeter number	Degrees of fundamental invariants
An	[3,3...,3]	...	...	n(n+1)/2	n + 1	n + 1	2, 3, 4, ..., n + 1
Bn	[4,3...,3]	...	...	n2	2n	2n − 1	2, 4, 6, ..., 2n
Cn			...			n + 1
Dn	[3,3,..31,1]	...	...	n(n-1)	2n − 2	2n − 2	n; 2, 4, 6, ..., 2n − 2
E6	[32,2,1]			36	12	12	2, 5, 6, 8, 9, 12
E7	[33,2,1]			63	18	18	2, 6, 8, 10, 12, 14, 18
E8	[34,2,1]			120	30	30	2, 8, 12, 14, 18, 20, 24, 30
F4	[3,4,3]			24	12	9	2, 6, 8, 12
G2	[6]			6	6	4	2, 6
H3	[5,3]		-	15	10		2, 6, 10
H4	[5,3,3]		-	60	30		2, 12, 20, 30
I2(p)	[p]		-	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 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.

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.

There are relations between the order g of the Coxeter group and the Coxeter number h:[3]

• [p]: 2h/g_p = 1
• [p,q]: 8/g_p,q = 2/p + 2/q -1
• [p,q,r]: 64h/g_p,q,r = 12 - p - 2q - r + 4/p + 4/r
• [p,q,r,s]: 16/g_p,q,r,s = 8/g_p,q,r + 8/g_q,r,s + 2/(ps) - 1/p - 1/q - 1/r - 1/s +1
• ...

For 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.

Distinct Coxeter elements correspond to orientations of the Coxeter diagram (i.e. to Dynkin quivers): the simple reflections corresponding to source vertices are written first, downstream vertices later, and sinks last. (The choice of order among non-adjacent vertices is irrelevant, since they correspond to commuting reflections.) A special choice is the alternating orientation, in which the simple reflections are partitioned into two sets of non-adjacent vertices, and all edges are oriented from the first to the second set.[4] The alternating orientation produces a special Coxeter element w satisfying ${\displaystyle w^{h/2}=w_{0}}$, where w₀ is the longest element, provided the Coxeter number h is even.

For ${\displaystyle A_{n-1}\cong S_{n}}$, the symmetric group on n elements, Coxeter elements are certain n-cycles: the product of simple reflections ${\displaystyle (1,2)(2,3)\cdots (n{-}1\,n)}$ is the Coxeter element ${\displaystyle (1,2,3,\dots ,n)}$.[5] For n even, the alternating orientation Coxeter element is:

${\displaystyle (1,2)(3,4)\cdots (2,3)(4,5)\cdots =(2,4,6,\ldots ,n{-}2,n,n{-}1,n{-}3,\ldots ,5,3,1).}$

There are ${\displaystyle 2^{n-2}}$ distinct Coxeter elements among the ${\displaystyle (n{-}1)!}$ n-cycles.

The dihedral group Dih_p is generated by two reflections that form an angle of ${\displaystyle 2\pi /2p}$, and thus the two Coxeter elements are their product in either order, which is a rotation by ${\displaystyle \pm 2\pi /p}$.

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[6] and is the plane on which P has eigenvalues e^2πi/h and e^−2πi/h = e^{2πi(h−1)/h}.[7] This plane was first systematically studied in (Coxeter 1948),[8] and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.[8]

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.[9] 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[9] 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.

In three dimensions, the symmetry of a regular polyhedron, {p,q}, with one directed Petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry S_h, [2⁺,h⁺], order h. Adding a mirror, the symmetry can be doubled to antiprismatic symmetry, D_hd, [2⁺,h], order 2h. In orthogonal 2D projection, this becomes dihedral symmetry, Dih_h, [h], order 2h.

Coxeter group	A3 Td	B3 Oh		H3 Ih
Regular polyhedron	{3,3}	{4,3}	{3,4}	{5,3}	{3,5}
Symmetry	S4, [2+,4+], (2×) D2d, [2+,4], (2*2)	S6, [2+,6+], (3×) D3d, [2+,6], (2*3)		S10, [2+,10+], (5×) D5d, [2+,10], (2*5)
Coxeter plane symmetry	Dih4, [4], (*4•)	Dih6, [6], (*6•)		Dih10, [10], (*10•)
Petrie polygons of the Platonic solids, showing 4-fold, 6-fold, and 10-fold symmetry.

In four dimensions, the symmetry of a regular polychoron, {p,q,r}, with one directed Petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry +¹/_h[C_h×C_h][10] (John H. Conway), (C_2h/C₁;C_2h/C₁) (#1', Patrick du Val (1964)[11]), order h.

Coxeter group	A4	B4		F4	H4
Regular polychoron	{3,3,3}	{3,3,4}	{4,3,3}	{3,4,3}	{5,3,3}	{3,3,5}
Symmetry	+1/5[C5×C5]	+1/8[C8×C8]		+1/12[C12×C12]	+1/30[C30×C30]
Coxeter plane symmetry	Dih5, [5], (*5•)	Dih8, [8], (*8•)		Dih12, [12], (*12•)	Dih30, [30], (*30•)
Petrie polygons of the regular 4D solids, showing 5-fold, 8-fold, 12-fold and 30-fold symmetry.

In five dimensions, the symmetry of a regular 5-polytope, {p,q,r,s}, with one directed Petrie polygon marked, is represented by the composite of 5 reflections.

Coxeter group	A5	B5		D5
Regular polyteron	{3,3,3,3}	{3,3,3,4}	{4,3,3,3}	h{4,3,3,3}
Coxeter plane symmetry	Dih6, [6], (*6•)	Dih10, [10], (*10•)		Dih8, [8], (*8•)

In dimensions 6 to 8 there are 3 exceptional Coxeter groups; one uniform polytope from each dimension represents the roots of the exceptional Lie groups E_n. The Coxeter elements are 12, 18 and 30 respectively.

E_n groups

Coxeter group	E6	E7	E8
Graph	122	231	421
Coxeter plane symmetry	Dih12, [12], (*12•)	Dih18, [18], (*18•)	Dih30, [30], (*30•)

• Longest element of a Coxeter group