Chamber operator

Chamber operator is a term used in the field of mathematics, specifically in the area of differential geometry. It refers to a mathematical operator that is used in the study of chambers in a Coxeter group.

Definition

In the context of a Coxeter group, a chamber operator is a function that maps each chamber to an adjacent chamber. The concept of a chamber operator is closely related to the concept of a reflection in a hyperplane, which is a fundamental operation in the theory of Coxeter groups.

Properties

Chamber operators have several important properties that make them useful in the study of Coxeter groups. For example, they are involutions, meaning that applying the same operator twice returns the original chamber. This property reflects the geometric fact that reflecting twice in the same hyperplane returns the original point.

Chamber operators also satisfy certain commutation relations, which are relationships between the results of applying different operators in different orders. These relations are encoded in the Coxeter diagram, which is a graphical representation of a Coxeter group.

Applications

Chamber operators are used in various areas of mathematics. In differential geometry, they are used in the study of symmetric spaces, which are spaces that are invariant under the action of a group of transformations. In combinatorics, they are used in the study of polytopes and tilings.

In addition, chamber operators play a crucial role in the theory of buildings, which are combinatorial structures that generalize the concept of a Coxeter group. Buildings can be thought of as "higher-dimensional" analogues of graphs, and chamber operators provide a way to navigate through these structures.

See also

• Coxeter group
• Reflection (mathematics)
• Hyperplane
• Involution (mathematics)
• Commutation relations
• Coxeter diagram
• Symmetric spaces
• Polytopes
• Tilings
• Buildings (mathematics)