Coxeter plane

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.

    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 number of roots divided by the rank. The number of reflections in the Coxeter group is half the number of roots.
    • The Coxeter number is the order of any Coxeter element;.
    • If the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi
    • The Coxeter number is 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 Coxeter 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] ... ... n + 1 n + 1 2, 3, 4, ..., n + 1
    Bn [4,3...,3] ... ... 2n 2n − 1 2, 4, 6, ..., 2n
    Cn ... n + 1
    Dn [3,3,..31,1] ... ... 2n − 2 2n − 2 n; 2, 4, 6, ..., 2n − 2
    E6 [32,2,1] 12 12 2, 5, 6, 8, 9, 12
    E7 [33,2,1] 18 18 2, 6, 8, 10, 12, 14, 18
    E8 [34,2,1] 30 30 2, 8, 12, 14, 18, 20, 24, 30
    F4 [3,4,3] 12 9 2, 6, 8, 12
    G2 [6] 6 4 2, 6
    H3 [5,3] - 10 2, 6, 10
    H4 [5,3,3] - 30 2, 12, 20, 30
    I2(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 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

    There are relations between group order, g, and the Coxeter number, h:

    • [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

    Coxeter elements of , considered as the symmetric group on n elements, are n-cycles: for simple reflections the adjacent transpositions , a Coxeter element is the n-cycle .

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

    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 ei/h and e−2πi/h = ei(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.

    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

    {3,3}

    {4,3}

    {3,4}

    {5,3}

    {3,5}
    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] (John H. Conway), (C2h/C1;C2h/C1) (#1', Patrick du Val (1964)), order h.

    Coxeter group A4, [3,3,3] BC4, [4,3,3] F4, [3,4,3] H4, [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[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

    {3,3,3,3}

    {3,3,3,4}

    {4,3,3,3}

    h{4,3,3,3}
    Coxeter plane
    symmetry
    Dih6, [6], (*6•) Dih10, [10], (*10•) Dih8, [8], (*8•)

    See also

    • Longest element of a Coxeter group

    Notes

    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 
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