such that for all , else . Writing , the associated Coxeter group is the group presented as having generators , , and relations
for all , whenever . In other words, is the order of (as is easily shown), and these orders determine the group.
A Coxeter group is usually more properly regarded as a group presentation rather than as an abstract group, but there is less than perfect consistency on this point in the literature. The are involutions that play the role of reflections generating the group.
Often Coxeter groups are specified by means of Coxeter diagrams. A Coxeter diagram associated with a Coxeter matrix is a multigraph whose vertices are indexed by , and with edges between distinct vertices . Coxeter diagrams are convenient visual aids; for example, involutions , commute precisely when there are no edges between and , and so the product of two Coxeter groups is specified by the disjoint union of their Coxeter diagrams.
Let be a finite dimensional real Euclidean space, and suppose is a finite group generated by linear reflections of (each pointwise fixing a linear hyperplane and preserving the Euclidean metric). Then is a Coxeter group. Such finite Coxeter groups are also called spherical Coxeter groups (being subgroups of isometry groups of Euclidean spheres).
In this case, each generating reflection is the mirror reflection specified by the fixed hyperplane , with for any orthogonal to . It is easy to check that is the order of , where is the dihedral angle between and . Moreover, the relations suffice to specify the structure of the group, roughly because the dihedral angles uniquely determine a hyperplane arrangement up to an element of the orthogonal group, and the hyperplane arrangement determines the group in kaleidoscopic fashion, using the hyperplanes as mirrors.
Finite reflection groups have been completely classified. Notice that if and are finite reflection groups on and respectively, then is a finite reflection group on . Such reflection groups are called reducible, and for the purposes of classification it suffices to consider just irreducible reflection groups. Also, if is a reflection group on , we can regard it as a reflection group on in a trivial way, but we ignore such inessential extensions in our descriptions below. Thus it will suffice to consider only irreducible, essential finite reflection groups. The dimension of the Euclidean space on which the group acts will be indicated by a subscript (with mild and fixable exceptions for and ).
Irreducible essential finite reflection groups fall into four infinite families , , , , together with a small number of exceptional groups:
The Coxeter matrices which specify these groups are often and traditionally encoded in the form of Coxeter diagrams, consisting of dots, and line segments between dots and if . The case where (no line segments) means and commute. Coxeter diagrams are highly convenient; for example, there are a number of “coincidences” where various Coxeter groups in different families A-I are isomorphic, and these coincidences are visually apparent by seeing that their Coxeter diagrams are isomorphic. Similarly, the Coxeter diagram of a reducible group is the disjoint union of the Coxeter diagrams for and separately, and so in the irreducible case we are only interested in connected Coxeter diagrams.
is the isometry group of a regular -simplex, and is identified with the symmetric group , where is identified with the permutation . We have , and if . The condition may be rewritten as a braid relation
since the are involutions.
is the isometry group of a regular -cube , and is identified with a wreath product . The generators may be given by where is the reflection through the hyperplane (i.e., swap the and coordinates), and is the reflection through the hyperplane . The Coxeter diagram looks like this:
so that is of order 4, but is otherwise of order 2 (i.e., and commute).
Remark: The distinction between and is not apparent at the level of Coxeter groups, but rather at the level of root systems, used to classify simple complex Lie algebras. In other words, in this case there are two distinct root systems which generate the same reflection group.
is the linear isometry group on the set of integral vectors in of length , of which there are many. It is not an isometry group of a regular solid, but it is a subgroup of of index 2. In this case there are involutions where is as described for the case , and swaps and negates the last two coordinates. The Coxeter diagram looks like this:
We see by examining the Coxeter diagrams that . The case on the other hand admits a symmetry of order 3, called triality?.
The dihedral group of order is the isometry group of a regular -gon in the plane, or of the set in . This is generated by two reflections: complex conjugation, and reflection through the hyperplane orthogonal to . The Coxeter diagram has edges between two vertices. There are coincidences , and .
There is also a coincidence (which allows us to elide over the description of the Coxeter group !). The distinction between them is again only at the level of root systems, where the root system of consists of the 12 vectors
There is a rich and fascinating literature on these structures; we confine ourselves only to a very succinct (and cryptic) description.
is the group of linear isometries of a root system consisting of vectors in such that
Its Coxeter diagram is:
is the group of linear isometries of the subset of roots used for whose first two coordinates are equal. Its Coxeter diagram is:
is the group of linear isometries of the subset of roots used for whose first three coordinates are equal. Its Coxeter diagram is:
The group is the isometry group of the 24-cell?, which is a regular polyhedron in consisting of the 8 unit quaternions , , , and the 16 unit quaternions given by where . (These 24 quaternions form a group under multiplication, and this group is isomorphic to .) The Coxeter diagram is:
See the discussion on above.
The group is the isometry group of the regular dodecahedron or of the regular icosahedron in , and is abstractly isomorphic to the group , having 120 elements. Its Coxeter diagram is:
The group is the isometry group of a regular polyhedron in known as the 120-cell, or of the dual polyhedron known as the 600-cell. Its Coxeter diagram is:
Equality in Coxeter groups is decidable. In other words, there is an algorithm which, given two words , in the generators , determines in finitely many steps whether belongs to the relator subgroup.
Consider the equivalence relation on words generated by the relation if is obtained by replacing an alternating substring of the form of length by the alternating substring of the same length. Clearly each equivalence class has only finitely many members since all the word-lengths are the same. Then say that is reduced if no member of its equivalence class contains a substring of the form . If is unreduced, then is -related to, hence in the same coset of the relator subgroup as, a which has such a substring , which in turn is in the coset of the word obtained by deleting , called a reduction of . The algorithm proceeds by enumerating all words in the -equivalence class of , passing to the first reduction that arises, and iterating this process until one finally obtains a reduced word ; this will be in the same relator coset as . Then, two reduced words are in the same relator coset if and only if they are -equivalent, and more generally two words , are in the same relator coset if and only if the algorithm applied to each produces two reduced words which are -equivalent.
Reduced-word expressions for a group element may be visualized as paths of minimal length from the identity to in the Cayley graph? given by the group presentation (here, a simple graph whose vertices are group elements, with an edge between and if for some generator ). Such reduced-word expressions are important in the study of buildings based on the Coxeter group, and also in the related study of BN-pairs?.
N. Bourbaki, Groupes et Algèbras de Lie, Chapitres 4-6, Masson, Paris (1981).
Kenneth Brown, Buildings, Springer Monographs in Mathematics, Springer 1989.