Warning: 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.
- The Coxeter number is the number of roots divided by the rank.
- 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||...||n + 1||n + 1||2, 3, 4, ..., n + 1|
|Bn||...||2n||2n − 1||2, 4, 6, ..., 2n|
|Cn||n + 1|
|Dn||...||2n − 2||2n − 2||n; 2, 4, 6, ..., 2n − 2|
|E6||12||12||2, 5, 6, 8, 9, 12|
|E7||18||18||2, 6, 8, 10, 12, 14, 18|
|E8||30||30||2, 8, 12, 14, 18, 20, 24, 30|
|F4||12||9||2, 6, 8, 12|
|G2 = I2(6)||6||4||2, 6|
|H3||10||2, 6, 10|
|H4||30||2, 12, 20, 30|
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 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 plane, below.
|This section requires expansion. (December 2008)|
The dihedral group Dihm is generated by two reflections that form an angle of , and thus their product is a rotation by .
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 e2πi/h and e−2πi/h = e2πi(h−1)/h. This plane was first systematically studied in (Coxeter 1948), and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.
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. 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 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.
- Coxeter, H. S. M. (1948), Regular Polytopes, Methuen and Co.
- Steinberg, Robert (June 1959), "Finite Reflection Groups", Transactions of the American Mathematical Society 91 (3): 493–504, ISSN 0002-9947, 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 0-273-08517-4
- Humphreys, James E. (1992), Reflection Groups and 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 Planes
- Stekolshchik, R. (2008), Notes on 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 Plane", Séminaire Lotharingien de Combinatoire B63b: 32