Coxeter Arrangement

Definition

A Coxeter arrangement is a specific type of hyperplane arrangement associated with a finite reflection group in Euclidean space. These arrangements are crucial for studying the geometric properties of reflection groups and their associated polytopes, as they reveal the combinatorial structure of these groups and their connections to other mathematical concepts.

5 Must Know Facts For Your Next Test

1. Coxeter arrangements can be defined in any dimension and are particularly important in the study of polytopes and their symmetries.
2. The intersection of the hyperplanes in a Coxeter arrangement can create a complex structure known as a complexified arrangement, which plays a significant role in combinatorial geometry.
3. Coxeter arrangements can be classified based on the types of reflection groups they are associated with, such as finite, affine, or spherical groups.
4. The arrangement's combinatorial properties are closely tied to the underlying geometry, influencing aspects like the number of regions created by the intersections.
5. These arrangements have applications beyond pure geometry, including areas like algebraic geometry and representation theory.

Review Questions

• How do Coxeter arrangements relate to reflection groups and what implications does this have for geometric properties?
  • Coxeter arrangements are intrinsically linked to reflection groups as they are formed from the hyperplanes corresponding to these groups. The reflection group determines how these hyperplanes interact and intersect, influencing the overall geometric structure. This relationship is crucial as it allows mathematicians to analyze the symmetries and combinatorial aspects of polytopes associated with these groups.
• Discuss the significance of hyperplanes in the context of Coxeter arrangements and how they contribute to understanding the arrangement's structure.
  • Hyperplanes are fundamental to Coxeter arrangements because they serve as the building blocks for defining the arrangement itself. Each hyperplane corresponds to a reflection in a finite reflection group, shaping the overall configuration. Understanding these hyperplanes allows for insights into how they intersect and form different regions within the arrangement, which is key for studying their geometric and combinatorial properties.
• Evaluate how Coxeter arrangements contribute to advancements in other fields of mathematics such as algebraic geometry and representation theory.
  • Coxeter arrangements extend their influence beyond discrete geometry into areas like algebraic geometry and representation theory by providing a framework for analyzing polynomial equations and symmetry properties. The intersection theory developed through Coxeter arrangements helps in understanding varieties defined by these polynomials. Moreover, their connections to representation theory arise from their ability to describe symmetries that relate to group actions on vector spaces, allowing for deeper explorations into character theory and module representations.

Related terms

Reflection Group: A group generated by reflections across hyperplanes in Euclidean space, which can be finite or infinite.

Hyperplane: A subspace of one dimension less than its ambient space, often used in geometry to define various arrangements.

Coxeter Diagram: A graphical representation of a Coxeter group, illustrating the relationships between the generators and their angles.