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In mathematics, the Coxeter
Coxeter
number h is the order of a Coxeter
Coxeter
element of an irreducible Coxeter
Coxeter
group. It is named after H.S.M. Coxeter.[1]

Contents

1 Definitions 2 Group order 3 Coxeter
Coxeter
elements 4 Coxeter
Coxeter
plane 5 See also 6 Notes 7 References

Definitions[edit] Note that this article assumes a finite Coxeter
Coxeter
group. For infinite Coxeter
Coxeter
groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter
Coxeter
number h of an irreducible root system. A Coxeter
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
Coxeter
number is the order of any Coxeter
Coxeter
element;. The Coxeter
Coxeter
number is 2m/n, where n is the rank, and m is the number of reflections. In the crystallographic case, m is the half the number of roots; and 2m+n is the dimension of the corresponding semisimple Lie algebra. If the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi. The Coxeter
Coxeter
number is the highest degree of a fundamental invariant of the Coxeter group
Coxeter group
acting on polynomials.

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

Coxeter
Coxeter
group Coxeter diagram Dynkin diagram Reflections m=nh/2[2] Coxeter
Coxeter
number h Dual Coxeter
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
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
Coxeter
element are the numbers e2πi(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 = e2πi/h, which is important in the Coxeter
Coxeter
plane, below. Group order[edit] There are relations between the order g of the Coxeter
Coxeter
group, and the Coxeter
Coxeter
number h:[3]

[p]: 2h/gp = 1 [p,q]: 8/gp,q = 2/p + 2/q -1 [p,q,r]: 64h/gp,q,r = 12 - p - 2q - r + 4/p + 4/r [p,q,r,s]: 16/gp,q,r,s = 8/gp,q,r + 8/gq,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
Coxeter
elements[edit]

This section needs expansion. You can help by adding to it. (December 2008)

Distinct Coxeter
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
Coxeter
element w satisfying

w

h

/

2

=

w

0

displaystyle w^ h/2 =w_ 0

, where w0 is the longest element, and we assume the Coxeter
Coxeter
number h is even. For

A

n − 1

S

n

displaystyle A_ n-1 cong S_ n

, the symmetric group on n elements, Coxeter
Coxeter
elements are certain n-cycles: the product of simple reflections

( 1 , 2 ) ( 2 , 3 ) ⋯ ( n

1

n )

displaystyle (1,2)(2,3)cdots (n - 1,n)

is the Coxeter
Coxeter
element

( 1 , 2 , 3 , … , n )

displaystyle (1,2,3,dots ,n)

.[5] For n even, the alternating orientation Coxeter
Coxeter
element is:

( 1 , 2 ) ( 3 , 4 ) ⋯ ( 2 , 3 ) ( 4 , 5 ) ⋯ = ( 2 , 4 , 6 , … , n

2 , n , n

1 , n

3 , … , 5 , 3 , 1 ) .

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

2

n − 2

displaystyle 2^ n-2

distinct Coxeter
Coxeter
elements among the

( n

1 ) !

displaystyle (n - 1)!

n-cycles. The dihedral group Dihp is generated by two reflections that form an angle of

2 π

/

2 p

displaystyle 2pi /2p

, and thus their product is a rotation by

2 π

/

p

displaystyle 2pi /p

. Coxeter
Coxeter
plane[edit]

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

For a given Coxeter
Coxeter
element w, there is a unique plane P on which w acts by rotation by 2π/h. This is called the Coxeter
Coxeter
plane[6] and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi(h−1)/h.[7] This plane was first systematically studied in ( Coxeter
Coxeter
1948),[8] and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.[8] The Coxeter
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
Coxeter
plane, yielding a Petrie polygon
Petrie polygon
with h-fold rotational symmetry.[9] For root systems, no root maps to zero, corresponding to the Coxeter
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
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 Sh, [2+,h+], order h. Adding a mirror, the symmetry can be doubled to antiprismatic symmetry, Dhd, [2+,h], order 2h. In orthogonal 2D projection, this becomes dihedral symmetry, Dihh, [h], order 2h.

Coxeter
Coxeter
group A3, [3,3] Td B3, [4,3] Oh H3, [5,3] Th

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
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
Petrie polygon
marked is a double rotation, defined as a composite of 4 reflections, with symmetry +1/h[Ch×Ch][10] (John H. Conway), (C2h/C1;C2h/C1) (#1', Patrick du Val (1964)[11]), order h.

Coxeter
Coxeter
group A4, [3,3,3] B4, [4,3,3] F4, [3,4,3] H4, [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[C5×C5] +1/8[C8×C8] +1/12[C12×C12] +1/30[C30×C30]

Coxeter
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
Petrie polygon
marked, is represented by the composite of 5 reflections.

Coxeter
Coxeter
group A5, [3,3,3,3] B5, [4,3,3,3] D5, [32,1,1]

Regular polyteron

3,3,3,3

3,3,3,4

4,3,3,3

h 4,3,3,3

Coxeter
Coxeter
plane symmetry Dih6, [6], (*6•) Dih10, [10], (*10•) Dih8, [8], (*8•)

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

En groups

Coxeter
Coxeter
group E6 E7 E8

Graph

122

231

421

Coxeter
Coxeter
plane symmetry Dih12, [12], (*12•) Dih18, [18], (*18•) Dih30, [30], (*30•)

See also[edit]

Longest element of a Coxeter
Coxeter
group

Notes[edit]

^ Coxeter, Harold Scott Macdonald; Chandler Davis; Erlich W. Ellers (2006), The Coxeter
Coxeter
Legacy: Reflections and Projections, AMS Bookstore, p. 112, ISBN 978-0-8218-3722-1  ^ 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
Coxeter
Planes and More Coxeter
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 978-1-56881-134-5 ^ Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.

References[edit]

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/S0002-9947-1959-0106428-2, ISSN 0002-9947, JSTOR 1993261  Hiller, Howard Geometry of Coxeter
Coxeter
groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. ISBN 0-273-08517-4 Humphreys, James E. (1992), Reflection Groups and Coxeter
Coxeter
Groups, Cambridge University Press, pp. 74–76 (Section 3.16, Coxeter Elements), ISBN 978-0-521-43613-7  Stembridge, John (April 9, 2007), Coxeter
Coxeter
Planes  Stekolshchik, R. (2008), Notes on Coxeter
Coxeter
Transformations and the McKay Correspondence, Springer Monographs in Mathematics, doi:10.1007/978-3-540-77398-3, ISBN 978-3-540-77398-6  Reading, Nathan (2010), "Noncrossing Partitions, Clusters and the Coxeter
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 Be

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