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.^{}
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 reflections in the Coxeter group is half the number of roots.
 The Coxeter number is the order of any Coxeter element;.
 If the highest root is ∑m_{i}α_{i} for simple roots α_{i}, then the Coxeter number is 1 + ∑m_{i}
 The Coxeter number is 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 Coxeter group acting on polynomials.
 The Coxeter number is given by the following table:
Coxeter group  Coxeter diagram 
Dynkin diagram 
Coxeter number h 
Dual Coxeter number  Degrees of fundamental invariants  

A_{n}  [3,3...,3]  ...  ...  n + 1  n + 1  2, 3, 4, ..., n + 1 
B_{n}  [4,3...,3]  ...  ...  2n  2n − 1  2, 4, 6, ..., 2n 
C_{n}  ...  n + 1  
D_{n}  [3,3,..3^{1,1}]  ...  ...  2n − 2  2n − 2  n; 2, 4, 6, ..., 2n − 2 
E_{6}  [3^{2,2,1}]  12  12  2, 5, 6, 8, 9, 12  
E_{7}  [3^{3,2,1}]  18  18  2, 6, 8, 10, 12, 14, 18  
E_{8}  [3^{4,2,1}]  30  30  2, 8, 12, 14, 18, 20, 24, 30  
F_{4}  [3,4,3]  12  9  2, 6, 8, 12  
G_{2}  [6]  6  4  2, 6  
H_{3}  [5,3]    10  2, 6, 10  
H_{4}  [5,3,3]    30  2, 12, 20, 30  
I_{2}(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.
Group order
There are relations between group order, g, and the Coxeter number, h:^{}
 [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
 ...
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
This section requires . (December 2008) 
Coxeter elements of , considered as the symmetric group on n elements, are ncycles: for simple reflections the adjacent transpositions , a Coxeter element is the ncycle .^{}
The dihedral group Dih_{m} is generated by two reflections that form an angle of , and thus their product is a rotation by .
Coxeter plane
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 e^{2πi/h} and e^{−2πi/h} = e^{2πi(h−1)/h}.^{} This plane was first systematically studied in (Coxeter 1948),^{} and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.^{}
The Coxeter plane is often used to draw diagrams of higherdimensional 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 hfold 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 w form hfold circular arrangements^{} and there is an empty center, as in the E_{8} 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  A_{3}, [3,3] T_{d} 
BC_{3}, [4,3] O_{h} 
H_{3}, [5,3] T_{h} 


Regular polyhedron 
{3,3} 
{4,3} 
{3,4} 
{5,3} 
{3,5} 
Symmetry  S_{4}, [2^{+},4^{+}], (2×) D_{2d}, [2^{+},4], (2*2) 
S_{6}, [2^{+},6^{+}], (3×) D_{3d}, [2^{+},6], (2*3) 
S_{10}, [2^{+},10^{+}], (5×) D_{5d}, [2^{+},10], (2*5) 

Coxeter plane symmetry 
Dih_{4}, [4], (*4•)  Dih_{6}, [6], (*6•)  Dih_{10}, [10], (*10•)  
Petrie polygons of the Platonic solids, showing 4fold, 6fold, and 10fold symmetry. 
In four dimension, 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}/C_{1};C_{2h}/C_{1}) (#1', Patrick du Val (1964)^{}), order h.
Coxeter group  A_{4}, [3,3,3]  BC_{4}, [4,3,3]  F_{4}, [3,4,3]  H_{4}, [5,3,3]  

Regular polychoron 
{3,3,3} 
{3,3,4} 
{4,3,3} 
{3,4,3} 
{5,3,3} 
{3,3,5} 

Symmetry  +^{1}/_{5}[C_{5}×C_{5}]  +^{1}/_{8}[C_{8}×C_{8}]  +^{1}/_{12}[C_{12}×C_{12}]  +^{1}/_{30}[C_{30}×C_{30}]  
Coxeter plane symmetry 
Dih_{5}, [5], (*5•)  Dih_{8}, [8], (*8•)  Dih_{12}, [12], (*12•)  Dih_{30}, [30], (*30•)  
Petrie polygons of the regular 4D solids, showing 5fold, 8fold, 12fold and 30fold symmetry. 
In five dimension, the symmetry of a regular polyteron, {p,q,r,s}, with one directed petrie polygon marked, is represented by the composite of 5 reflections.
Coxeter group  A_{5}, [3,3,3,3]  BC_{5}, [4,3,3,3]  D_{5}, [3^{2,1,1}]  

Regular polyteron 
{3,3,3,3} 
{3,3,3,4} 
{4,3,3,3} 
h{4,3,3,3} 
Coxeter plane symmetry 
Dih_{6}, [6], (*6•)  Dih_{10}, [10], (*10•)  Dih_{8}, [8], (*8•) 
See also
 Longest element of a Coxeter group
Notes
References
 Coxeter, H. S. M. (1948), Regular Polytopes, Methuen and Co.
 Steinberg, R. (June 1959), "Finite Reflection Groups", Transactions of the American Mathematical Society 91 (3): 493–504, doi:10.1090/S00029947195901064282, ISSN 00029947, JSTOR 1993261
 Hiller, Howard Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.London, 1982. iv+213 pp. ISBN 0273085174
 Humphreys, James E. (1992), Reflection Groups and Coxeter Groups, Cambridge University Press, pp. 74–76 (Section 3.16, Coxeter Elements), ISBN 9780521436137
 Stembridge, John (April 9, 2007), Coxeter Planes
 Stekolshchik, R. (2008), Notes on Coxeter Transformations and the McKay Correspondence, Springer Monographs in Mathematics, doi:10.1007/9783540773983, ISBN 9783540773986
 Reading, Nathan (2010), "Noncrossing Partitions, Clusters and the Coxeter Plane", Séminaire Lotharingien de Combinatoire B63b: 32