Coxeter Diagram

Definition

A Coxeter diagram is a graphical representation used to describe a Coxeter group, which is a group defined by reflections across hyperplanes in a Euclidean space. The diagram consists of nodes representing the generators of the group and edges indicating the angles between these reflections, providing a compact way to visualize complex relationships within the group. Understanding Coxeter diagrams is essential when exploring root systems and their symmetries, as they help illustrate the connections between different root spaces and weights.

5 Must Know Facts For Your Next Test

1. Coxeter diagrams can represent finite and infinite groups, with finite diagrams corresponding to finite reflection groups and finite root systems.
2. The edges in a Coxeter diagram may be labeled with integers indicating the order of the angles between the generators, where an edge without a label implies an angle of 120 degrees.
3. Each connected component of a Coxeter diagram corresponds to a specific type of root system, such as A_n or D_n, providing insight into the symmetries of these systems.
4. The classification of Coxeter groups can be done through their diagrams, which enables mathematicians to connect algebraic properties with geometric interpretations.
5. Coxeter diagrams play a critical role in the study of Weyl groups, which are essential in understanding representations of Lie algebras and their associated weight spaces.

Review Questions

• How do Coxeter diagrams represent relationships between different generators in a Coxeter group?
  • Coxeter diagrams visually illustrate the relationships between generators by using nodes for each generator and edges to signify the angles between the corresponding reflections. The presence and labeling of edges provide insight into how these reflections interact, allowing for an understanding of the symmetries within the associated root system. This representation helps clarify the structure and properties of the group generated by these reflections.
• Discuss the significance of edges in a Coxeter diagram and what they convey about the angles between reflections.
  • In a Coxeter diagram, edges carry crucial information about the angles between reflections corresponding to the generators. A labeled edge indicates a specific integer that denotes how many times one reflection must be applied before returning to its original position relative to another generator. Unlabeled edges imply standard angles (e.g., 120 degrees), establishing rules for how reflections combine geometrically. This understanding is key when analyzing root systems and their symmetries.
• Evaluate how Coxeter diagrams contribute to our understanding of root systems and weight spaces in algebra.
  • Coxeter diagrams significantly enhance our understanding of root systems and weight spaces by providing a clear visual framework for analyzing symmetries and relationships within these structures. They allow mathematicians to classify different types of root systems based on their geometric properties and identify connections to their corresponding Weyl groups. By bridging the gap between algebraic concepts and geometric interpretations, Coxeter diagrams facilitate deeper insights into representation theory and its applications in various areas of mathematics.

Related terms

Root System: A collection of roots in a vector space that satisfies certain symmetrical properties, crucial for understanding the structure of Lie algebras and their representations.

Reflection Group: A group generated by reflections across hyperplanes, closely related to Coxeter groups and useful in analyzing symmetry in geometric contexts.

Dynkin Diagram: A type of graph used to classify semisimple Lie algebras, similar in purpose to Coxeter diagrams but specifically focused on algebraic structures.