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

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:

$$\underbrace {\rho _{i}\rho _{j}\rho _{i}\rho _{j}\dots } _{k}=\underbrace {\rho _{j}\rho _{i}\rho _{j}\rho _{i}\dots } _{k}$$

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:

$$\left(\rho _{i}\rho _{j}\right)^{k}=\varepsilon$$

Coxeter matrix

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

$$\left(\rho _{i}\rho _{j}\right)^{k}=\varepsilon$$

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

Main article: Coxeter-Dynkin diagram

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

• The Coxeter matrix $\left[{\begin{matrix}1&5&2\\5&1&3\\2&3&1\end{matrix}}\right]$ has the diagram
• The Coxeter matrix $\left[{\begin{matrix}1&4&2&2\\4&1&3&2\\2&3&1&3\\2&2&3&1\end{matrix}}\right]$ has the diagram
• The Coxeter matrix $\left[{\begin{matrix}1&4&2&3\\4&1&3&2\\2&3&1&4\\3&2&4&1\end{matrix}}\right]$ has the diagram

Linear Coxeter groups

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 that group is a linear Coxeter group.

See also

• 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

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