Coxeter group

From Groupprops
Revision as of 07:22, 15 May 2015 by Vipul (talk | contribs) (Metaproperties)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Coxeter group, all facts related to Coxeter group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

Symbol-free definition

A group is termed a Coxeter group if it can be equipped with a finite presentation given by the following relations:

  • The square of every generator
  • For some of the pairwise products of the generators, a certain power of that pairwise product

A presentation of this kind is termed a Coxeter presentation. Often, the term Coxeter group is used for the group along with a specific choice of Coxeter presentation.

Definition with symbols

A group G is termed a Coxeter group if it can be equipped with a finite presentation with generators s_i and relations:

  • s_i^2 = 1
  • (s_is_j)^{m_{ij}} = 1 where m_{ij} is a symmetric function of i and j (for distinct i and j)

Alternatively we can consider a symmetric matrix m_{ij} with the diagonal entries being 1 and simply require that for each i and j (not necessarily distinct) (s_is_j)^{m_{ij}} = 1. Note that we allow the entries m_{ij} to be 0.

Such a presentation is termed a Coxeter presentation and the matrix of m_{ij}s is termed a Coxeter matrix. Often, the term Coxeter group is used for a Coxeter group along with a specific choice of Coxeter presentation.

Particular cases

Number of generators Form of Coxeter matrix Common name for the group Comment
2 \begin{pmatrix} 1 & a \\ a & 1 \\\end{pmatrix} dihedral group of degree a, order 2a.
2 \begin{pmatrix} 1 & 2 \\ 2 & 1 \\\end{pmatrix} Klein four-group
2 \begin{pmatrix} 1 & 3 \\ 3 & 1 \\\end{pmatrix} symmetric group of degree three
2 \begin{pmatrix} 1 & 4 \\ 4 & 1 \\\end{pmatrix} dihedral group of order eight
2 \begin{pmatrix} 1 & 5 \\ 5 & 1 \\\end{pmatrix} dihedral group of order ten
2 \begin{pmatrix} 1 & 6 \\ 6 & 1 \\\end{pmatrix} dihedral group of order twelve
2 \begin{pmatrix} 1 & 8 \\ 8 & 1 \\\end{pmatrix} dihedral group of order sixteen
3 \begin{pmatrix} 1 & l & m \\ l & 1 & n \\ m & n & 1 \\\end{pmatrix} triangle group with parameters (l,m,n)
3 \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \\\end{pmatrix} elementary abelian group of order eight
3 \begin{pmatrix} 1 & l & 2 \\ l & 1 & 2 \\ 2 & 2 & 1 \\\end{pmatrix} Direct product of dihedral group of degree l (order 2l) and cyclic group of order two
3 \begin{pmatrix} 1 & 3 & 3 \\ 3 & 1 & 2 \\ 3 & 2 & 1 \\\end{pmatrix} symmetric group of degree four
3 \begin{pmatrix} 1 & 4 & 3 \\ 4 & 1 & 2 \\ 3 & 2 & 1 \\\end{pmatrix} direct product of S4 and Z2
3 \begin{pmatrix} 1 & 5 & 3 \\ 5 & 1 & 3 \\ 3 & 2 & 1 \\\end{pmatrix} direct product of A5 and Z2
3 \begin{pmatrix} 1 & 7 & 3 \\ 7 & 1 & 2 \\ 3 & 2 & 1 \\\end{pmatrix} (7,3,2)-triangle group this group is infinite.
n 1s on diagonal, 3s on superdiagonal and subdiagonal, 2s elsewhere. symmetric group of degree n + 1.

Metaproperties

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

A direct product of Coxeter groups is a Coxeter group. The Coxeter matrix for the direct product is obtained by taking the block concatenation of the Coxeter matrices for the individual groups and then replacing the off-diagonal zero blocks by blocks with all entries equal to 2.