Coxeter groups are abstract groups that can be represented by reflections across hyperplanes in a Euclidean space, and they play a fundamental role in the study of symmetries and geometric structures. They are defined by a set of generators subject to relations that can be encoded in a Coxeter matrix or diagram, allowing for the classification and analysis of symmetrical properties in various dimensions.

Coxeter Groups

Abstract groups

Reflections across hyperplanes

Euclidean space is a mathematical construct that generalizes the properties of two-dimensional and three-dimensional spaces to any number of dimensions, characterized by the notions of distance and angle. It serves as the foundational setting for classical geometry and is defined by a coordinate system where the distance between points is given by the Euclidean distance formula.

Euclidean Space

Study of symmetries

Geometric structures are mathematical frameworks that study the properties and relations of points, lines, surfaces, and solids in space. They are foundational in fields such as topology, algebraic geometry, and differential geometry, providing essential insights into the nature of shapes and spaces.

Geometric Structures

Set of generators

A Coxeter matrix is a square matrix associated with a Coxeter group, where each entry represents the order of the product of two generators of the group. It is symmetric, with diagonal elements equal to one, and off-diagonal elements that are either integers greater than one or infinity, indicating the absence of a relation between the corresponding generators.

Coxeter Matrix

A Coxeter diagram is a graphical representation of a Coxeter group, which encodes the group's abstract structure by illustrating the relations between its generating reflections. Each node in the diagram represents a generator, and edges, often labeled with integers, denote the order of the product of the reflections corresponding to the connected nodes.

Concepts

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