Coxeter element

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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]

Definitions[edit]

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 ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi
  • 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
An [3,3...,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png Dyn-node.pngDyn-3.pngDyn-node.pngDyn-3.png...Dyn-3.pngDyn-node.pngDyn-3.pngDyn-node.png n + 1 n + 1 2, 3, 4, ..., n + 1
Bn [4,3...,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png Dyn-node.pngDyn-4a.pngDyn-node.pngDyn-3.png...Dyn-3.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.png 2n 2n − 1 2, 4, 6, ..., 2n
Cn Dyn-node.pngDyn-4b.pngDyn-node.pngDyn-3.png...Dyn-3.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.png n + 1
Dn [3,3,..31,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png Dyn-branch1.pngDyn-node.pngDyn-3.png...Dyn-3.pngDyn-node.pngDyn-3.pngDyn-node.png 2n − 2 2n − 2 n; 2, 4, 6, ..., 2n − 2
E6 [32,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Dyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.png 12 12 2, 5, 6, 8, 9, 12
E7 [33,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Dyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.png 18 18 2, 6, 8, 10, 12, 14, 18
E8 [34,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Dyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.png 30 30 2, 8, 12, 14, 18, 20, 24, 30
F4 [3,4,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png Dyn-node.pngDyn-3.pngDyn-node.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.png
Dyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-node.pngDyn-3.pngDyn-node.png
12 9 2, 6, 8, 12
G2 [6] CDel node.pngCDel 6.pngCDel node.png Dyn-node.pngDyn-6a.pngDyn-node.png
Dyn-node.pngDyn-6b.pngDyn-node.png
6 4 2, 6
H3 [5,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png - 10 2, 6, 10
H4 [5,3,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png - 30 2, 12, 20, 30
I2(p) [p] CDel node.pngCDel p.pngCDel node.png - 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 ei(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 = ei/h, which is important in the Coxeter plane, below.

Group order[edit]

There are relations between group order, g, and the Coxeter number, h:[2]

  • [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 elements[edit]

Coxeter elements of , considered as the symmetric group on n elements, are n-cycles: for simple reflections the adjacent transpositions , a Coxeter element is the n-cycle .[3]

The dihedral group Dihm is generated by two reflections that form an angle of , and thus their product is a rotation by .

Coxeter plane[edit]

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[4] and is the plane on which P has eigenvalues ei/h and e−2πi/h = ei(h−1)/h.[5] This plane was first systematically studied in (Coxeter 1948),[6] and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.[6]

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.[7] 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[7] 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 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 group A3, [3,3]
Td
B3, [4,3]
Oh
H3, [5,3]
Th
Regular
polyhedron
3-simplex t0.svg
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3-cube t0.svg
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
3-cube t2.svg
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Dodecahedron t0 H3.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Icosahedron t0 H3.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
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[Ch×Ch][8] (John H. Conway), (C2h/C1;C2h/C1) (#1', Patrick du Val (1964)[9]), order h.

Coxeter group A4, [3,3,3] B4, [4,3,3] F4, [3,4,3] H4, [5,3,3]
Regular
polychoron
4-simplex t0.svg
{3,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4-orthoplex.svg
{3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
4-cube graph.svg
{4,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
24-cell t0 F4.svg
{3,4,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
120-cell graph H4.svg
{5,3,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
600-cell graph H4.svg
{3,3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
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, [3,3,3,3] B5, [4,3,3,3] D5, [32,1,1]
Regular
polyteron
5-simplex t0.svg
{3,3,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-orthoplex.svg
{3,3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube graph.svg
{4,3,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube t0 D5.svg
h{4,3,3,3}
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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 En Exceptional lie groups. The Coxeter elements are 12, 18 and 30 respectively.

En groups
Coxeter group E6 E7 E8
Graph Up 1 22 t0 E6.svg
122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Gosset 2 31 polytope.svg
231
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
E8Petrie.svg
421
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Coxeter plane
symmetry
Dih12, [12], (*12•) Dih18, [18], (*18•) Dih30, [30], (*30•)

See also[edit]

Notes[edit]

  1. ^ Coxeter, Harold Scott Macdonald; Chandler Davis; Erlich W. Ellers (2006), The Coxeter Legacy: Reflections and Projections, AMS Bookstore, p. 112, ISBN 978-0-8218-3722-1 
  2. ^ Regular polytopes, p. 233
  3. ^ (Humphreys 1992, p. 75)
  4. ^ Coxeter Planes and More Coxeter Planes John Stembridge
  5. ^ (Humphreys 1992, Section 3.17, "Action on a Plane", pp. 76–78)
  6. ^ a b (Reading 2010, p. 2)
  7. ^ a b (Stembridge 2007)
  8. ^ On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
  9. ^ Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.

References[edit]