Coxeter element

From Wikipedia, the free encyclopedia
  (Redirected from Coxeter plane)
Jump to: navigation, search
Not to be confused with Longest element of a Coxeter group.

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group, hence also of a root system or its Weyl 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 mirrors in the Coxeter group is half the number of roots.
  • The Coxeter number is the order of a Coxeter element; note that conjugate elements have the same order.
  • 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 Weyl 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 12 9 2, 6, 8, 12
G2 [6,3] CDel node.pngCDel 6.pngCDel node.png Dyn-node.pngDyn-6a.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 A_{n-1} \cong S_n, considered as the symmetric group on n elements, are n-cycles: for simple reflections the adjacent transpositions (1,2), (2,3), \dots, a Coxeter element is the n-cycle (1,2,3,\dots, n).[3]

The dihedral group Dihm is generated by two reflections that form an angle of 2\pi/2m, and thus their product is a rotation by 2\pi/m.

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

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.[6] 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[6] 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
BC3, [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 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[Ch×Ch][7] (John H. Conway), (C2h/C1;C2h/C1) (#1', Patrick du Val (1964)[8]), order h.

Coxeter group A4, [3,3,3] BC4, [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 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 A5, [3,3,3,3] BC5, [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•)

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. ^ (Humphreys 1992, Section 3.17, "Action on a Plane", pp. 76–78)
  5. ^ a b (Reading 2010, p. 2)
  6. ^ a b (Stembridge 2007)
  7. ^ On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
  8. ^ Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.

References[edit]