Coxeter element

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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 number h Dual Coxeter number Degrees of fundamental invariants
An CDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png n + 1 n + 1 2, 3, 4, ..., n + 1
Bn CDel node.pngCDel 4.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png 2n 2n − 1 2, 4, 6, ..., 2n
Cn n + 1
Dn CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.png...CDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 2n − 2 2n − 2 n; 2, 4, 6, ..., 2n − 2
E6 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 12 12 2, 5, 6, 8, 9, 12
E7 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 18 18 2, 6, 8, 10, 12, 14, 18
E8 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 30 30 2, 8, 12, 14, 18, 20, 24, 30
F4 CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 12 9 2, 6, 8, 12
G2 = I2(6) CDel node.pngCDel 6.pngCDel node.png 6 4 2, 6
H3 CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 10 2, 6, 10
H4 CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 30 2, 12, 20, 30
I2(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.

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).[2]

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.[3] This plane was first systematically studied in (Coxeter 1948),[4] and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.[4]

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.[5] 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[5] 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.

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. ^ (Humphreys 1992, p. 75)
  3. ^ (Humphreys 1992, Section 3.17, "Action on a Plane", pp. 76–78)
  4. ^ a b (Reading 2010, p. 2)
  5. ^ a b (Stembridge 2007)

References[edit]