Coxeter Diagram

Definition

A Coxeter diagram is a graphical representation that encodes information about a Coxeter group, which is generated by reflections in hyperplanes. These diagrams help illustrate the relationships between the generators of the group and can be used to classify different types of geometric structures, such as root systems and reflection groups. Each node represents a generator, and the edges denote the angles between them, which can indicate how the reflections relate to each other.

5 Must Know Facts For Your Next Test

1. Coxeter diagrams can represent both finite and infinite groups, making them versatile in studying geometric properties.
2. The edges in a Coxeter diagram are often labeled with integers that indicate the angles between the reflecting hyperplanes, where a label of 2 means orthogonal reflections.
3. Different orientations and connections in a Coxeter diagram can correspond to distinct geometric structures and symmetries, such as regular polytopes.
4. Coxeter diagrams play a crucial role in understanding how various algebraic structures, including Lie algebras, can be classified.
5. For any given Coxeter diagram, one can derive a Cartan matrix which is essential in classifying the corresponding Lie algebra associated with the diagram.

Review Questions

• How does a Coxeter diagram represent the relationships between generators in a Coxeter group?
  • A Coxeter diagram visually represents the generators of a Coxeter group as nodes, while the edges connecting these nodes indicate the angles between their corresponding reflections. If two generators are connected by an edge labeled with an integer, this value indicates how they interact geometrically. This relationship is crucial for understanding the structure of the group and how it acts in geometric settings.
• Compare and contrast Coxeter diagrams and Dynkin diagrams in terms of their applications in geometry and algebra.
  • Both Coxeter diagrams and Dynkin diagrams serve to encode relationships within algebraic structures, but they differ in their specific applications. Coxeter diagrams focus on reflecting hyperplanes and their interactions within various geometric contexts, while Dynkin diagrams are primarily concerned with root systems in Lie algebras. Although both types of diagrams aid in classifying algebraic entities, Dynkin diagrams have stricter conventions that limit their representations compared to Coxeter diagrams.
• Evaluate the significance of Coxeter diagrams in classifying Lie algebras and their associated root systems.
  • Coxeter diagrams are integral to classifying Lie algebras as they provide a systematic way to represent the relationships among roots within a root system. By associating each diagram with a Cartan matrix, one can derive information about the algebra's structure and properties. This classification not only aids in understanding the symmetries inherent within these algebras but also connects to broader mathematical themes such as representation theory and geometry. The ability to visualize these relationships helps mathematicians explore deeper connections between various mathematical objects.

Related terms

Coxeter Group: A group generated by reflections across hyperplanes in a Euclidean space, defined by a finite set of generators.

Dynkin Diagram: A specific type of graph that represents root systems, similar to Coxeter diagrams but with different conventions for encoding relationships.

Root System: A set of vectors in Euclidean space that satisfy certain symmetrical properties, used in the study of Lie algebras and algebraic groups.