Coxeter element

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 of an irreducible root system. • The Coxeter number is the order of any Coxeter element;. • The Coxeter number is {{tmath|\tfrac{2m}{n},}} where is the rank, and is the number of reflections. In the crystallographic case, is half the number of roots; and is the dimension of the corresponding semisimple Lie algebra. • If the highest root is \sum m_i \alpha_i for simple roots , then the Coxeter number is 1 + \sum 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: 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 is a degree of a fundamental invariant then so is . The eigenvalues of a Coxeter element are the numbers e^{2\pi i\frac{m-1}{h}} as runs through the degrees of the fundamental invariants. Since this starts with , these include the primitive th root of unity, \zeta_h = e^{2\pi i\frac{1}{h}}, which is important in the Coxeter plane, below. The dual Coxeter number is 1 plus the sum of the coefficients of simple roots in the highest short root of the dual root system. == Group order==

There are relations between the order of the Coxeter group and the Coxeter number : \begin{align} {} [p]:& \quad \frac{2h}{g_p} = 1 \\[4pt] [p,q]:& \quad \frac{8}{g_{p,q}} = \frac{2}{p} + \frac{2}{q} -1 \\[4pt] [p,q,r]:& \quad \frac{64h}{g_{p,q,r}} = 12 - p - 2q - r + \frac{4}{p} + \frac{4}{r} \\[4pt] [p,q,r,s]:& \quad \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 \\[4pt] \vdots \qquad & \qquad \vdots \end{align} For example, has : \begin{align} &\frac{64 \times 30}{g_{3,3,5}} = 12 - 3 - 6 - 5 + \frac{4}{3} + \frac{4}{5} = \frac{2}{15}, \\[4pt] &\therefore g_{3,3,5} = \frac{1920\times 15}{2} = 960 \times 15 = 14400. \end{align} ==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 satisfying w^{h/2}= w_0, where is the longest element, provided the Coxeter number is even. For A_{n-1} \cong S_n, the symmetric group on elements, Coxeter elements are certain -cycles: the product of simple reflections (1,2) (2,3) \cdots (n-1,n) is the Coxeter element (1,2,3,\dots, n). For even, the alternating orientation Coxeter element is: (1,2)(3,4)\cdots (2,3)(4,5) \cdots = (2,4,6,\ldots,n{-}2,n, n{-}1,n{-}3,\ldots,5,3,1). There are 2^{n-2} distinct Coxeter elements among the (n{-}1)! -cycles. The dihedral group is generated by two reflections that form an angle of \tfrac{2\pi}{2p}, and thus the two Coxeter elements are their product in either order, which is a rotation by \pm \tfrac{2\pi}{p}. ==Coxeter plane==

For a given Coxeter element , there is a unique plane on which acts by rotation by {{tmath|\tfrac{2\pi}{h}.}} This is called the Coxeter plane and is the plane on which has eigenvalues e^{2\pi i\frac{1}{h}} and e^{-2\pi i\frac{1}{h}} = e^{2\pi i\frac{h-1}{h}}. This plane was first systematically studied in , and subsequently used in to provide uniform proofs about properties of Coxeter elements. 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 form -fold circular arrangements (John H. Conway), (#1', Patrick du Val (1964)), order . In five dimensions, the symmetry of a regular 5-polytope, {{math|{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 . The Coxeter elements are 12, 18 and 30 respectively. ==See also==