Coxeter element
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.^{[1]}
Contents
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 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 ∑m_{i}α_{i} for simple roots α_{i}, then the Coxeter number is 1 + ∑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=nh/2^{[2]} 
Coxeter number h 
Dual Coxeter number  Degrees of fundamental invariants  

A_{n}  [3,3...,3]  ...  ...  n(n+1)/2  n + 1  n + 1  2, 3, 4, ..., n + 1 
B_{n}  [4,3...,3]  ...  ...  n^{2}  2n  2n − 1  2, 4, 6, ..., 2n 
C_{n}  ...  n + 1  
D_{n}  [3,3,..3^{1,1}]  ...  ...  n(n1)  2n − 2  2n − 2  n; 2, 4, 6, ..., 2n − 2 
E_{6}  [3^{2,2,1}]  36  12  12  2, 5, 6, 8, 9, 12  
E_{7}  [3^{3,2,1}]  63  18  18  2, 6, 8, 10, 12, 14, 18  
E_{8}  [3^{4,2,1}]  120  30  30  2, 8, 12, 14, 18, 20, 24, 30  
F_{4}  [3,4,3] 

24  12  9  2, 6, 8, 12  
G_{2}  [6] 

6  6  4  2, 6  
H_{3}  [5,3]    15  10  2, 6, 10  
H_{4}  [5,3,3]    60  30  2, 12, 20, 30  
I_{2}(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.
Group order
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
 ...
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
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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 nonadjacent 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 nonadjacent vertices, and all edges are oriented from the first to the second set.^{[4]} The alternating orientation produces a special Coxeter element w satisfying , where w_{0} is the longest element, and we assume the Coxeter number h is even.
For , the symmetric group on n elements, Coxeter elements are certain ncycles: the product of simple reflections is the Coxeter element .^{[5]} For n even, the alternating orientation Coxeter element is:
There are distinct Coxeter elements among the ncycles.
The dihedral group Dih_{p} 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^{[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 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.^{[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 hfold circular arrangements^{[9]} 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} 
B_{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 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}/C_{1};C_{2h}/C_{1}) (#1', Patrick du Val (1964)^{[11]}), order h.
Coxeter group  A_{4}, [3,3,3]  B_{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 dimensions, the symmetry of a regular 5polytope, {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]  B_{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•) 
In dimensions 6 to 8 there are 3 exceptional Coxeter groups, one uniform polytope from each dimension represents the roots of the E_{n} Exceptional lie groups. The Coxeter elements are 12, 18 and 30 respectively.
Coxeter group  E6  E7  E8 

Graph 
1_{22} 
2_{31} 
4_{21} 
Coxeter plane symmetry 
Dih_{12}, [12], (*12•)  Dih_{18}, [18], (*18•)  Dih_{30}, [30], (*30•) 
See also
Notes
 ^ Coxeter, Harold Scott Macdonald; Chandler Davis; Erlich W. Ellers (2006), The Coxeter Legacy: Reflections and Projections, AMS Bookstore, p. 112, ISBN 9780821837221
 ^ Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61
 ^ Regular polytopes, p. 233
 ^ George Lusztig, Introduction to Quantum Groups, Birkhauser (2010)
 ^ (Humphreys 1992, p. 75)
 ^ Coxeter Planes and More Coxeter Planes John Stembridge
 ^ (Humphreys 1992, Section 3.17, "Action on a Plane", pp. 76–78)
 ^ ^{a} ^{b} (Reading 2010, p. 2)
 ^ ^{a} ^{b} (Stembridge 2007)
 ^ On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 9781568811345
 ^ Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
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
 Bernšteĭn, I. N.; Gelʹfand, I. M.; Ponomarev, V. A., "Coxeter functors, and Gabriel's theorem" (Russian), Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33. Translation on Bernstein's website.