Coxeter complexes

Definition

Coxeter complexes are combinatorial structures that arise from the study of reflection groups and their associated geometry. They are constructed from the vertices of a simplicial complex, where the edges represent reflections across hyperplanes corresponding to the group elements. This relationship links Coxeter complexes to toric varieties, as they can provide a way to understand the geometry of these varieties through the lens of combinatorial data.

5 Must Know Facts For Your Next Test

1. Coxeter complexes can be represented as graphs, where vertices correspond to the simple roots of the associated root system.
2. The structure of a Coxeter complex provides insight into the combinatorial properties of the underlying reflection group.
3. These complexes play a significant role in understanding the link between geometry and combinatorial topology in toric varieties.
4. The connections between Coxeter complexes and toric varieties often manifest through their respective fans and polyhedral structures.
5. Coxeter complexes also serve as an important tool for studying equidimensional decompositions of algebraic varieties.

Review Questions

• How do Coxeter complexes relate to reflection groups and what implications does this have for their combinatorial structure?
  • Coxeter complexes are closely tied to reflection groups, as they are built from the geometry defined by these groups. The edges in a Coxeter complex represent reflections across hyperplanes related to group elements, and this establishes a combinatorial structure that reflects the symmetry properties of the associated geometric objects. Understanding this relationship helps in analyzing how different elements of the group interact within the complex, revealing insights about both geometric symmetries and algebraic properties.
• Discuss the importance of Coxeter complexes in understanding toric varieties, focusing on how they contribute to the geometry and combinatorial data of these varieties.
  • Coxeter complexes play a pivotal role in understanding toric varieties by providing a combinatorial framework to analyze their geometry. The relationships between the simplices of Coxeter complexes and the cones in fans associated with toric varieties allow mathematicians to translate geometric problems into combinatorial terms. This interplay enhances our understanding of how toric varieties can be realized geometrically and highlights their structural properties derived from reflection groups.
• Evaluate how Coxeter complexes can facilitate the exploration of more complex algebraic structures and their relationships with geometric properties.
  • Coxeter complexes serve as a powerful tool for exploring complex algebraic structures by linking them with their geometric counterparts. By providing a combinatorial perspective, these complexes allow mathematicians to analyze intricate relationships between different types of varieties and their underlying symmetries. This evaluation is crucial for advancing theories in algebraic geometry, as it enables deeper investigations into how geometric properties can inform the understanding of algebraic structures, leading to new insights and potential applications in higher-dimensional geometry.

Related terms

Toric Varieties: Algebraic varieties that can be described using combinatorial data associated with fans, which are collections of cones in a vector space.

Reflection Group: A group generated by reflections across hyperplanes, which act on a vector space and can be associated with geometric symmetries.

Simplicial Complex: A set of simplices (points, line segments, triangles, etc.) that fit together in a specific way to form a topological space.