Coxeter number
Encyclopedia
In mathematics
, the Coxeter number h is the order
of a Coxeter element of an irreducible Coxeter group
, hence also of a root system
or its Weyl group
. It is named after H.S.M. Coxeter.
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 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 e2πi(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 = e2πi/h, which is important in the Coxeter plane, below.
, 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 
.
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 solid
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Coxeter number h is the order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
of a Coxeter element of an irreducible Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
, hence also of a root system
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
or its Weyl group
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
. It is named after H.S.M. Coxeter.
Definitions
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
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
.
- The Coxeter number is the number of roots divided by the rank.
- The Coxeter number is the order of a Coxeter element; note that conjugate elements have the same order.
- If the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi
- The dimension of the corresponding Lie algebraLie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
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 Weyl group acting on polynomials.
- The Coxeter number is given by the following table:
| Coxeter group | Coxeter number h | Dual Coxeter number | Degrees of fundamental invariants | |
|---|---|---|---|---|
| An | ... | n + 1 | n + 1 | 2, 3, 4, ..., n + 1 |
| Bn | ... | 2n | 2n − 1 | 2, 4, 6, ..., 2n |
| Cn | n + 1 | |||
| Dn | ... | 2n − 2 | 2n − 2 | n; 2, 4, 6, ..., 2n − 2 |
| E6 | 12 | 12 | 2, 5, 6, 8, 9, 12 | |
| E7 | 18 | 18 | 2, 6, 8, 10, 12, 14, 18 | |
| E8 | 30 | 30 | 2, 8, 12, 14, 18, 20, 24, 30 | |
| F4 | 12 | 9 | 2, 6, 8, 12 | |
| G2 = I2(6) | 6 | 4 | 2, 6 | |
| H3 | 10 | 2, 6, 10 | ||
| H4 | 30 | 2, 12, 20, 30 | ||
| I2(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 e2πi(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 = e2πi/h, which is important in the Coxeter plane, below.
Coxeter elements
Coxeter elements of, considered as the symmetric groupSymmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
on n elements, are n-cycles: for simple reflections the adjacent transpositions
, a Coxeter element is the n-cycle .The dihedral group
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
Dihm 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 and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi(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
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...
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 solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s.