Coxeter diagram

Definition

A Coxeter diagram is a graphical representation of a Coxeter group, which is a type of group defined by reflections in Euclidean space. The diagram consists of nodes representing generators of the group and edges indicating the relations between these generators, with specific labels on the edges that provide information about the angles between the corresponding reflections. This visualization helps in understanding the underlying geometric structures and symmetries associated with root systems and weights.

5 Must Know Facts For Your Next Test

1. Coxeter diagrams can encode different types of geometric symmetries, including those found in regular polyhedra and higher-dimensional shapes.
2. The nodes in a Coxeter diagram are typically labeled with integers or omitted entirely, depending on the type of relationship they represent among generators.
3. Coxeter diagrams can be used to classify types of root systems, including those corresponding to classical Lie groups like $A_n$, $B_n$, $C_n$, and $D_n$.
4. The structure of the Coxeter diagram provides insight into the angles between the reflections, helping to determine how the associated root system can be constructed geometrically.
5. The study of Coxeter diagrams extends beyond geometry into areas such as algebra, combinatorics, and even theoretical physics through their connections with symmetry and representation theory.

Review Questions

• How do Coxeter diagrams facilitate the understanding of relationships between generators in Coxeter groups?
  • Coxeter diagrams provide a clear visual representation of the generators in a Coxeter group through nodes and edges. Each node corresponds to a generator, while the edges indicate the relationships between them, such as the angles formed by their reflections. This graphical approach makes it easier to analyze complex interactions among generators, leading to better comprehension of the group’s structure.
• Discuss the importance of edge labels in a Coxeter diagram and what they signify about the geometric properties of root systems.
  • Edge labels in a Coxeter diagram convey crucial information regarding the angles between corresponding reflections. For example, an edge labeled with an integer represents the angle formed by two reflections being related, while an absence of a label implies orthogonality. Understanding these labels allows for deeper insights into the geometric properties and symmetries present within root systems, ultimately aiding in their classification.
• Evaluate how Coxeter diagrams contribute to both algebraic and geometric aspects of representation theory, particularly in relation to Lie algebras.
  • Coxeter diagrams serve as a bridge between algebraic structures and geometric interpretations within representation theory. They encapsulate vital information about root systems associated with Lie algebras while simultaneously providing insights into their symmetrical properties. By examining these diagrams, one can discern patterns in weights and roots that reveal deeper algebraic relationships, facilitating advancements in understanding both representation theory and its applications across mathematics and physics.

Related terms

Coxeter group: A mathematical structure generated by reflections in Euclidean space, characterized by a set of relations between these reflections.

Root system: A finite set of vectors (roots) in a Euclidean space that satisfy certain symmetrical properties and play a crucial role in the classification of Lie algebras.

Weights: Elements associated with representations of Lie algebras, which can be thought of as points in a vector space that correspond to certain linear functionals.