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.

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 Template:Mvar of an irreducible root system.

• The Coxeter number is the order of any Coxeter element;.
• The Coxeter number is Template:Tmath where Template:Mvar is the rank, and Template:Mvar is the number of reflections. In the crystallographic case, Template:Mvar is half the number of roots; and 2m+n is the dimension of the corresponding semisimple Lie algebra.
• If the highest root is $\sum m_i{\alpha }_i$ for simple roots Template:Mvar, 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.

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

Coxeter group		Coxeter diagram	Dynkin diagram	Reflections $m=\frac{nh}{2}$	Coxeter number Template:Mvar	Dual Coxeter number	Degrees of fundamental invariants
An	[3,3...,3]	Template:CDD...Template:CDD	Template:Dynkin...Template:Dynkin	$\frac{n(n+1)}{2}$	n + 1	n + 1	2, 3, 4, ..., n + 1
Bn	[4,3...,3]	Template:CDD...Template:CDD	Template:Dynkin...Template:Dynkin	n2	2n	2n − 1	2, 4, 6, ..., 2n
Cn			Template:Dynkin...Template:Dynkin			n + 1
Dn	[3,3,...31,1]	Template:CDD...Template:CDD	Template:Dynkin...Template:Dynkin	n(n − 1)	2n − 2	2n − 2	n; 2, 4, 6, ..., 2n − 2
E6	[32,2,1]	Template:CDD	Template:Dynkin2	36	12	12	2, 5, 6, 8, 9, 12
E7	[33,2,1]	Template:CDD	Template:Dynkin2	63	18	18	2, 6, 8, 10, 12, 14, 18
E8	[34,2,1]	Template:CDD	Template:Dynkin2	120	30	30	2, 8, 12, 14, 18, 20, 24, 30
F4	[3,4,3]	Template:CDD	Template:Dynkin Template:Dynkin	24	12	9	2, 6, 8, 12
G2	[6]	Template:CDD	Template:Dynkin Template:Dynkin	6	6	4	2, 6
H3	[5,3]	Template:CDD	-	15	10		2, 6, 10
H4	[5,3,3]	Template:CDD	-	60	30		2, 12, 20, 30
I2(p)	[p]	Template:CDD	-	Template:Mvar	Template:Mvar		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 Template:Mvar 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 Template:Mvar runs through the degrees of the fundamental invariants. Since this starts with m = 2, these include the [[primitive root of unity|primitive Template:Mvarth 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.

Group order

There are relations between the order Template:Mvar of the Coxeter group and the Coxeter number Template:Mvar: $$\begin{array}{rl}[p]:& \frac{2h}{g_p}=1\\ [p,q]:& \frac{8}{g_{p,q}}=\frac{2}{p}+\frac{2}{q}-1\\ [p,q,r]:& \frac{64h}{g_{p,q,r}}=12-p-2q-r+\frac{4}{p}+\frac{4}{r}\\ [p,q,r,s]:& \frac{16}{g_{p,q,r,s}}=\frac{8}{g_{p,q,r}}+\frac{8}{g_{q,r,s}}+\frac{2}{ps}-\frac{1}{p}-\frac{1}{q}-\frac{1}{r}-\frac{1}{s}+1\\ ⋮& ⋮\end{array}$$

For example, [3,3,5] has h = 30: $$\begin{array}{rl}& \frac{64\times 30}{g_{3,3,5}}=12-3-6-5+\frac{4}{3}+\frac{4}{5}=\frac{2}{15},\\ & \therefore g_{3,3,5}=\frac{1920\times 15}{2}=960\times 15=14400.\end{array}$$

Coxeter elements

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. The alternating orientation produces a special Coxeter element Template:Mvar satisfying $w^{h/2}=w_0,$ where w₀ is the longest element, provided the Coxeter number Template:Mvar is even.

For $A_{n-1}\cong S_n,$ the symmetric group on Template:Mvar elements, Coxeter elements are certain Template:Mvar-cycles: the product of simple reflections $(1,2)(2,3)\cdots (n-1,n)$ is the Coxeter element $(1,2,3,\dots ,n)$. For Template:Mvar even, the alternating orientation Coxeter element is: $$(1,2)(3,4)\cdots (2,3)(4,5)\cdots =(2,4,6,\dots ,n-2,n,n-1,n-3,\dots ,5,3,1).$$ There are $2^{n-2}$ distinct Coxeter elements among the $(n-1)!$ Template:Mvar-cycles.

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

Coxeter plane

File:E8Petrie.svg

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

For a given Coxeter element Template:Mvar, there is a unique plane Template:Mvar on which Template:Mvar acts by rotation by Template:Tmath This is called the Coxeter plane and is the plane on which Template:Mvar has eigenvalues $e^{2\pi i\frac{1}{h}}$ and $e^{-2\pi i\frac{1}{h}}=e^{2\pi i\frac{h-1}{h}}.$ This plane was first systematically studied in , and subsequently used in to provide uniform proofs about properties of Coxeter elements.

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 Template:Mvar-fold rotational symmetry. 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 Template:Mvar form Template:Mvar-fold circular arrangements 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 Template:Mvar. 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	File:3-simplex t0.svg Tetrahedron {3,3} Template:CDD	File:3-cube t0.svg Cube {4,3} Template:CDD	File:3-cube t2.svg Octahedron {3,4} Template:CDD	File:Dodecahedron H3 projection.svg Dodecahedron {5,3} Template:CDD	File:Icosahedron H3 projection.svg Icosahedron {3,5} Template:CDD
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] (John H. Conway), (C_2h/C₁;C_2h/C₁) (#1', Patrick du Val (1964)), order Template:Mvar.

Coxeter group	A4	B4		F4	H4
Regular polychoron	File:4-simplex t0.svg 5-cell {3,3,3} Template:CDD	File:4-orthoplex.svg 16-cell {3,3,4} Template:CDD	File:4-cube graph.svg Tesseract {4,3,3} Template:CDD	File:24-cell t0 F4.svg 24-cell {3,4,3} Template:CDD	File:120-cell graph H4.svg 120-cell {5,3,3} Template:CDD	File:600-cell graph H4.svg 600-cell {3,3,5} Template:CDD
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	File:5-simplex t0.svg 5-simplex {3,3,3,3} Template:CDD	File:5-orthoplex.svg 5-orthoplex {3,3,3,4} Template:CDD	File:5-cube graph.svg 5-cube {4,3,3,3} Template:CDD	File:5-demicube t0 D5.svg 5-demicube h{4,3,3,3} Template:CDD
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	File:Up 1 22 t0 E6.svg 122 Template:CDD	File:Gosset 2 31 polytope.svg 231 Template:CDD	File:E8Petrie.svg 421 Template:CDD
Coxeter plane symmetry	Dih12, [12], (*12•)	Dih18, [18], (*18•)	Dih30, [30], (*30•)

See also

• Longest element of a Coxeter group

References

• Hiller, Howard Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. Template:ISBN
• Bernšteĭn, I. N.; Gelʹfand, I. M.; Ponomarev, V. A., "Coxeter functors, and Gabriel's theorem" (Russian), Uspekhi Mat. Nauk 28 (1973), no. 2(170), 19–33. Translation on Bernstein's website.