Coxeter element

Group order

There are relations between the order g of the Coxeter group and the Coxeter number h: $${\displaystyle \begin{array}{cc}[p]:& \frac{2h}{g_p}=1\\ [p,q]:& \frac{8}{g_{p,q}}=\frac{2}{p}+\frac{2}{q}-1\\ [p,q,r]:& \frac{64h}{g_{p,q,r}}=12-p-2q-r+\frac{4}{p}+\frac{4}{r}\\ [p,q,r,s]:& \frac{16}{g_{p,q,r,s}}=\frac{8}{g_{p,q,r}}+\frac{8}{g_{q,r,s}}+\frac{2}{ps}-\frac{1}{p}-\frac{1}{q}-\frac{1}{r}-\frac{1}{s}+1\\ ⋮& ⋮\end{array}}$$

For example, [3,3,5] has h = 30: $${\displaystyle \begin{array}{cc}& \frac{64\times 30}{g_{3,3,5}}=12-3-6-5+\frac{4}{3}+\frac{4}{5}=\frac{2}{15},\\ & \therefore g_{3,3,5}=\frac{1920\times 15}{2}=960\times 15=14400.\end{array}}$$

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. The alternating orientation produces a special Coxeter element w satisfying ${\displaystyle w^{h/2}=w_0,}$ where w₀ is the longest element, provided the Coxeter number h is even.

For ${\displaystyle A_{n-1}\cong S_n,}$ the symmetric group on n elements, Coxeter elements are certain n-cycles: the product of simple reflections ${\displaystyle (1,2)(2,3)\cdots (n-1,n)}$ is the Coxeter element ${\displaystyle (1,2,3,\dots ,n)}$. For n even, the alternating orientation Coxeter element is: $${\displaystyle (1,2)(3,4)\cdots (2,3)(4,5)\cdots =(2,4,6,\dots ,n-2,n,n-1,n-3,\dots ,5,3,1).}$$ There are ${\displaystyle 2^{n-2}}$ distinct Coxeter elements among the ${\displaystyle (n-1)!}$ n-cycles.

The dihedral group Dih_p is generated by two reflections that form an angle of ${\displaystyle \frac{2\pi }{2p},}$ and thus the two Coxeter elements are their product in either order, which is a rotation by ${\displaystyle \pm \frac{2\pi }{p}.}$

Coxeter plane

For a given Coxeter element w, there is a unique plane P on which w acts by rotation by ⁠${\displaystyle \frac{2\pi }{h}.}$⁠ This is called the Coxeter plane and is the plane on which P has eigenvalues ${\displaystyle e^{2\pi i\frac{1}{h}}}$ and ${\displaystyle e^{-2\pi i\frac{1}{h}}=e^{2\pi i\frac{h-1}{h}}.}$ 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 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.

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] (John H. Conway), (C_2h/C₁;C_2h/C₁) (#1', Patrick du Val (1964)), order h.

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.

In dimensions 6 to 8 there are 3 exceptional Coxeter groups; one uniform polytope from each dimension represents the roots of the exceptional Lie groups E_n. The Coxeter elements are 12, 18 and 30 respectively.