Coxeter element explained

Coxeter element should not be confused with Longest element of a Coxeter group.

In mathematics, a Coxeter element is an element of an irreducible Coxeter group which 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. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups.

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 of an irreducible root system.

• The Coxeter number is the order of any Coxeter element;.
• The Coxeter number is 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:

Coxeter group		Coxeter diagram	Dynkin diagram	Reflections m = \tfrac{nh}{ 2 } ^[1]	Coxeter number	Dual Coxeter number	Degrees of fundamental invariants
		...	...	(n(n +1 ))/(2)
		...	...
			...
		...	...
			-
			-
			-

The invariants of the Coxeter group acting on polynomials form a polynomial algebrawhose 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 (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 (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 :^[2]

For example, has :

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.^[3] 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 ) … (n -1 ,n) is the Coxeter element ( 1 , 2 , 3 , ... , n) . For even, the alternating orientation Coxeter element is:$$(1,2)(3,4)\backslash cdots\; (2,3)(4,5)\; \backslash cdots\; =\; (2,4,6,\backslash ldots,n2,n,\; n1,n3,\backslash ldots,5,3,1).$$ 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}{ 2 p}, 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 This is called the Coxeter plane^[4] and is the plane on which has eigenvalues

e ^{2 \pi i (1)/(h)} and e ^{-2 \pi i (1)/(h)} = e ^{2 \pi i (h -1)/(h)} . This plane was first systematically studied in, and subsequently used in 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 -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 form -fold circular arrangements and there is an empty center, as in the 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, with one directed Petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry,, order . Adding a mirror, the symmetry can be doubled to antiprismatic symmetry,,, order . In orthogonal 2D projection, this becomes dihedral symmetry,,, order .

Coxeter group	width=150
Regular polyhedron	Tetrahedron	Cube	Octahedron	Dodecahedron	Icosahedron
Symmetry
Coxeter plane symmetry
Petrie polygons of the Platonic solids, showing 4-fold, 6-fold, and 10-fold symmetry.

In four dimensions, the symmetry of a regular polychoron, with one directed Petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry ^[5] (John H. Conway), (#1', Patrick du Val (1964)^[6]), order .

Coxeter group
Regular polychoron	5-cell	16-cell	Tesseract	24-cell	120-cell	600-cell
Symmetry
Coxeter plane symmetry
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, 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.

Notes and References

1. [Coxeter]
2. Regular polytopes, p. 233
3. George Lusztig, Introduction to Quantum Groups, Birkhauser (2010)
4. http://www.math.lsa.umich.edu/~jrs/coxplane.html Coxeter Planes
5. On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith
6. Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.