Coxeter group

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Within group theory, Coxeter groups are a kind of abstract group which have a close relationship with groups generated by geometric reflections. All Coxeter groups have a corresponding Coxeter-Dynkin diagram.

Definition[edit | edit source]

Coxeter groups can be defined in terms of a group presentation. A rank n  Coxeter group is a group presentation generated by n  involutions (i.e. for each generator, ρ i , we have the relation ρ 2
= ε 
), plus additional relations of the form:

where ρ i  and ρ j  are generators. Relations of this form are called braid relations.

Because the generators are involutions these relations can also be written as:

Coxeter matrix[edit | edit source]

Since each relation in the presentation of a Coxeter group (including the involution requirement on the generators) is of the form:

It is determined by the two generators it relates and the power k .

Thus the entire Coxeter group with n  generators can be represented as an n  by n  symmetric matrix called the Coxeter matrix. Each row and column corresponds to a generator of the group, with the value of M i,j  being the power k  in the relation on the generators ρ i  and ρ j . If there is no relation between to generators we put   as their power in the matrix.

The Coxeter matrix is always symmetric, and its diagonal consists of 1s.

Coxeter-Dynkin diagrams[edit | edit source]

We can adapt the Coxeter matrix to a more concise format. We use the Coxeter matrix as the adjacency matrix for a edge-labeled graph. That is for an n ×n  Coxeter matrix, M , we create a graph with n  nodes connecting node i  and j  with an edge labeled M i,j  ignoring the cases where i  = j . Since the usually the majority of the entries in a given matrix are 2, we remove all edges labeled 2 to greatly simplify the resulting diagram. By convention we usually draw the diagram leaving edges labeled 3 unlabeled.

Examples[edit | edit source]

  • The Coxeter matrix has the diagram
  • The Coxeter matrix has the diagram
  • The Coxeter matrix has the diagram

Linear Coxeter groups[edit | edit source]

A Coxeter group is linear if there exists an indexing of its generators such that all pairs of non-adjacent generators commute.

These groups are called linear because their Coxeter diagram can be drawn as a single line.

Non-adjacent distinguished generators of a regular polytope must commute, thus if the symmetry group of a regular polytope is a Coxeter group[note 1] that group is a linear Coxeter group.

See also[edit | edit source]

External links[edit | edit source]

Notes[edit | edit source]

  1. This is the case for all planar polytopes, and many but not all abstract polytopes