Coxeter element

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

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 Coxeter number is the order of a Coxeter element; note that conjugate elements have the same order.
• If the highest root is ∑m_iα_i for simple roots α_i, then the Coxeter number is 1 + ∑m_i
• 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
I2(p)		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 e^{2πi(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, ζ_h = e^2πi/h, which is important in the Coxeter plane, below.

Coxeter elements

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 Dih_m is generated by two reflections that form an angle of 2π / 2m, and thus their product is a rotation by 2π / m.

Coxeter plane

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 e^2πi/h and e^−2πi/h = e^{2πi(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.

• 

  Petrie polygons of the Platonic solids, showing 4-fold, 6-fold, and 10-fold symmetry, corresponding to the Coxeter lengths of A₃, BC₃, and H₃.

See also

• Longest element of a Coxeter group

Notes

1. ^ Coxeter, Harold Scott Macdonald; Chandler Davis, Erlich W. Ellers (2006), The Coxeter Legacy: Reflections and Projections, AMS Bookstore, p. 112, ISBN 0821837222, 9780821837221, http://books.google.com/?id=cKpBGcqpspIC&pg=PA107&dq=%22Coxeter+number%22+%22Donald+Coxeter%22 
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

• 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  edit
• 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 52143613 7, http://books.google.com/?id=ODfjmOeNLMUC 
• Stembridge, John (April 9, 2007), Coxeter Planes, http://www.math.lsa.umich.edu/~jrs/coxplane.html 
• 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  edit
• Reading, Nathan (2010), "Noncrossing Partitions, Clusters and the Coxeter Plane", Séminaire Lotharingien de Combinatoire B63b: 32, http://www.emis.de/journals/SLC/wpapers/s63reading.html