A Coxeter group is a mathematical structure that describes symmetries in geometric shapes. It is defined by a set of generators and relations, which specify how these generators can be combined. These groups are named after the mathematician H.S.M. Coxeter, who studied their properties and applications in geometry. Coxeter groups can be represented using diagrams called Coxeter diagrams, which visually depict the relationships between generators. They play a significant role in various areas of mathematics, including group theory, geometry, and combinatorics, and are particularly useful in understanding the symmetries of regular polytopes and higher-dimensional shapes.