An affine Coxeter group is an extension of a finite Coxeter group that acts on an affine space, typically realized as a reflection group generated by reflections across hyperplanes. These groups are crucial in the study of symmetries and have applications in areas such as algebraic geometry, representation theory, and mathematical physics.

Affine Coxeter Group

Finite Coxeter group

An affine space is a geometric structure that generalizes the properties of Euclidean spaces, allowing for the definition of points and vectors without a fixed origin. It is characterized by the ability to perform vector addition and scalar multiplication while maintaining the concept of parallelism and affine transformations.

Affine Space

A reflection group is a group generated by reflections, which are isometries of a Euclidean space that fix a hyperplane pointwise. These groups play a crucial role in the study of symmetries and are foundational in understanding the geometric structure of spaces, particularly in the classification of finite groups and the theory of Coxeter groups.

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