CoxeterGruppen

Coxeter Gruppen, also known as Coxeter Groups, are a class of mathematical objects that play a significant role in various branches of mathematics, particularly in geometry and group theory.

In 1934, the mathematician Harold Scott MacDonald Coxeter introduced the concept of Coxeter graphs, which are used to define and classify these groups. A Coxeter group is a group that can be generated by a set of elements satisfying a specific set of relations, which are derived from the graph associated with the group.

The relations in a Coxeter group are defined by a Coxeter matrix, which is a square matrix containing information about the length of words in the presentation of the group. The Coxeter matrix is used to determine the order of the elements in the group and the length of words in the group.

Coxeter groups have a rich algebraic structure, including a presentation by generators and relations, and a presentation matrix. These groups have been widely studied and have many applications in mathematics and physics. They have connections to geometry, particularly in the study of crystallography and symmetry groups, as well as in Lie theory and representation theory.

Some examples of Coxeter groups include the symmetric group, the dihedral groups, and the affine Weyl groups. Coxeter groups also have a rich history, with key contributions from mathematicians such as Coxeter, Ewing, and Tits, among others. Today, Coxeter groups continue to be an active area of research in mathematics, with ongoing applications in computer science, physics, and materials science.