Descriptions of Coxeter Groups

A Coxeter system is a group G with finite generating set S={s₁, ..., s_n}, defined by the power relations s_i²=1 for i=1, ..., n and braid relations

s_is_js_i ... = s_js_is_j ... for i, j=1, ..., n with i<j, where each side of this relation has length m_ij≥2. Although traditionally m_ij=∞ signifies that the corresponding relation is omitted, for technical reasons, we use m_ij=0 instead. Set m_ji=m_ij and m_ii=1. The group G is called a Coxeter group and S is called the set of Coxeter generators. Since every group in Magma has a preferred generating set, no distinction is made between a Coxeter system and its Coxeter group.

Due to the importance and ubiquity of Coxeter groups, a number of different ways of describing these groups and their reflections have been developed. Functions for manipulating these descriptions are described in Chapter COXETER SYSTEMS.

Coxeter groups are usually described by a Coxeter matrix M=(m_ij)_{i, j=1}ⁿ, or by a Coxeter graph with vertices 1, ..., n and an edge connecting i and j labeled by m_ij whenever m_ij≥3.

Coxeter systems are mainly important because they provide presentations for the real reflection groups. A Cartan matrix describes a particular reflection representation of a Coxeter group. In certain cases, such a representation can be described by an integer-labelled digraph, called the Dynkin digraph (this is equivalent to a Dynkin diagram, but we have modified the definition for technical reasons).

For finite and affine Coxeter groups, the naming system due to Cartan is also used. Hyperbolic Coxeter groups of degree larger than 3 are numbered. [Next][Prev] [Right] [____] [Up] [Index] [Root]