# Coxeter group

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

i = 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]

- Shephard groups, a generalization of linear Coxeter groups.
- C string groups, quotients of linear Coxeter groups and the automorphism groups of abstract regular polytopes.
- Artin-Tits groups, groups generated with braid relations closely related to Coxeter groups.

## External links[edit | edit source]

- Wikipedia contributors. "Coxeter group".
- Weisstein, Eric W. "Coxeter Group" at MathWorld.
- nLab contributors. "Coxeter group" on nLab.

## Notes[edit | edit source]

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

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