Coxeter element
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
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 order of any Coxeter element;.
- The Coxeter number is 2m/n, where n is the rank, and m is the number of reflections. In the crystallographic case, m is half the number of roots; and 2m+n is the dimension of the corresponding semisimple Lie algebra.
- If the highest root is ∑m_{i}α_{i} for simple roots α_{i}, then the Coxeter number is 1 + ∑m_{i}.
- The Coxeter number is the highest degree of a fundamental invariant of the Coxeter group acting on polynomials.
The Coxeter number for each Dynkin type is given in the following table:
Coxeter group | Coxeter diagram |
Dynkin diagram |
Reflections m=nh/2[2] |
Coxeter number h |
Dual Coxeter number | Degrees of fundamental invariants | |
---|---|---|---|---|---|---|---|
A_{n} | [3,3...,3] | n(n+1)/2 | n + 1 | n + 1 | 2, 3, 4, ..., n + 1 | ||
B_{n} | [4,3...,3] | n^{2} | 2n | 2n − 1 | 2, 4, 6, ..., 2n | ||
C_{n} | n + 1 | ||||||
D_{n} | [3,3,..3^{1,1}] | n(n-1) | 2n − 2 | 2n − 2 | n; 2, 4, 6, ..., 2n − 2 | ||
E_{6} | [3^{2,2,1}] | 36 | 12 | 12 | 2, 5, 6, 8, 9, 12 | ||
E_{7} | [3^{3,2,1}] | 63 | 18 | 18 | 2, 6, 8, 10, 12, 14, 18 | ||
E_{8} | [3^{4,2,1}] | 120 | 30 | 30 | 2, 8, 12, 14, 18, 20, 24, 30 | ||
F_{4} | [3,4,3] | 24 | 12 | 9 | 2, 6, 8, 12 | ||
G_{2} | [6] | 6 | 6 | 4 | 2, 6 | ||
H_{3} | [5,3] | - | 15 | 10 | 2, 6, 10 | ||
H_{4} | [5,3,3] | - | 60 | 30 | 2, 12, 20, 30 | ||
I_{2}(p) | [p] | - | 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.
Group order
There are relations between the order g of the Coxeter group, and the Coxeter number h:[3]
- [p]: 2h/g_{p} = 1
- [p,q]: 8/g_{p,q} = 2/p + 2/q -1
- [p,q,r]: 64h/g_{p,q,r} = 12 - p - 2q - r + 4/p + 4/r
- [p,q,r,s]: 16/g_{p,q,r,s} = 8/g_{p,q,r} + 8/g_{q,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
Distinct Coxeter elements correspond to orientations of the Coxeter diagram (i.e. to Dynkin quivers): the simple reflections corresponding to source vertices are written first, downstream vertices later, and sinks last. (The choice of order among non-adjacent vertices is irrelevant, since they correspond to commuting reflections.) A special choice is the alternating orientation, in which the simple reflections are partitioned into two sets of non-adjacent vertices, and all edges are oriented from the first to the second set.[4] The alternating orientation produces a special Coxeter element w satisfying , where w_{0} is the longest element, and we assume the Coxeter number h is even.
For , the symmetric group on n elements, Coxeter elements are certain n-cycles: the product of simple reflections is the Coxeter element .[5] For n even, the alternating orientation Coxeter element is:
There are distinct Coxeter elements among the n-cycles.
The dihedral group Dih_{p} is generated by two reflections that form an angle of , and thus their product is a rotation by .
Coxeter plane
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[6] and is the plane on which P has eigenvalues e^{2πi/h} and e^{−2πi/h} = e^{2πi(h−1)/h}.[7] This plane was first systematically studied in (Coxeter 1948),[8] and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.[8]
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.[9] 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[9] and there is an empty center, as in the E_{8} 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 S_{h}, [2^{+},h^{+}], order h. Adding a mirror, the symmetry can be doubled to antiprismatic symmetry, D_{hd}, [2^{+},h], order 2h. In orthogonal 2D projection, this becomes dihedral symmetry, Dih_{h}, [h], order 2h.
Coxeter group | A_{3}, [3,3] T_{d} |
B_{3}, [4,3] O_{h} |
H_{3}, [5,3] T_{h} | ||
---|---|---|---|---|---|
Regular polyhedron |
{3,3} |
{4,3} |
{3,4} |
{5,3} |
{3,5} |
Symmetry | S_{4}, [2^{+},4^{+}], (2×) D_{2d}, [2^{+},4], (2*2) |
S_{6}, [2^{+},6^{+}], (3×) D_{3d}, [2^{+},6], (2*3) |
S_{10}, [2^{+},10^{+}], (5×) D_{5d}, [2^{+},10], (2*5) | ||
Coxeter plane symmetry |
Dih_{4}, [4], (*4•) | Dih_{6}, [6], (*6•) | Dih_{10}, [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}[C_{h}×C_{h}][10] (John H. Conway), (C_{2h}/C_{1};C_{2h}/C_{1}) (#1', Patrick du Val (1964)[11]), order h.
Coxeter group | A_{4}, [3,3,3] | B_{4}, [4,3,3] | F_{4}, [3,4,3] | H_{4}, [5,3,3] | ||
---|---|---|---|---|---|---|
Regular polychoron |
{3,3,3} |
{3,3,4} |
{4,3,3} |
{3,4,3} |
{5,3,3} |
{3,3,5} |
Symmetry | +^{1}/_{5}[C_{5}×C_{5}] | +^{1}/_{8}[C_{8}×C_{8}] | +^{1}/_{12}[C_{12}×C_{12}] | +^{1}/_{30}[C_{30}×C_{30}] | ||
Coxeter plane symmetry |
Dih_{5}, [5], (*5•) | Dih_{8}, [8], (*8•) | Dih_{12}, [12], (*12•) | Dih_{30}, [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 | A_{5}, [3,3,3,3] | B_{5}, [4,3,3,3] | D_{5}, [3^{2,1,1}] | |
---|---|---|---|---|
Regular polyteron |
{3,3,3,3} |
{3,3,3,4} |
{4,3,3,3} |
h{4,3,3,3} |
Coxeter plane symmetry |
Dih_{6}, [6], (*6•) | Dih_{10}, [10], (*10•) | Dih_{8}, [8], (*8•) |
In dimensions 6 to 8 there are 3 exceptional Coxeter groups, one uniform polytope from each dimension represents the roots of the E_{n} Exceptional lie groups. The Coxeter elements are 12, 18 and 30 respectively.
Coxeter group | E6 | E7 | E8 |
---|---|---|---|
Graph | 1_{22} |
2_{31} |
4_{21} |
Coxeter plane symmetry |
Dih_{12}, [12], (*12•) | Dih_{18}, [18], (*18•) | Dih_{30}, [30], (*30•) |
See also
Notes
- 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
- Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61
- Regular polytopes, p. 233
- George Lusztig, Introduction to Quantum Groups, Birkhauser (2010)
- (Humphreys 1992, p. 75)
- Coxeter Planes and More Coxeter Planes John Stembridge
- (Humphreys 1992, Section 3.17, "Action on a Plane", pp. 76–78)
- (Reading 2010, p. 2)
- (Stembridge 2007)
- On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
- Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
References
- Coxeter, H. S. M. (1948), Regular Polytopes, Methuen and Co.
- Steinberg, R. (June 1959), "Finite Reflection Groups", Transactions of the American Mathematical Society, 91 (3): 493–504, doi:10.1090/S0002-9947-1959-0106428-2, 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
- Bernšteĭn, I. N.; Gelʹfand, I. M.; Ponomarev, V. A., "Coxeter functors, and Gabriel's theorem" (Russian), Uspekhi Mat. Nauk 28 (1973), no. 2(170), 19–33. Translation on Bernstein's website.