User:Cullinane/Reflection group

Reflection groups "are groups (acting on a finite dimensional vector space) generated by reflections: elements that fix a hyperplane pointwise. They include the Weyl and Coxeter groups, complex reflection groups..., and reflection groups over arbitrary fields. Topics include arrangements of hyperplanes, invariant theory,  modular reflection groups, regular polytopes, Hecke algebras, coinvariant algebras, Coxeter groups, Shephard Groups, and Braid groups."

-- Anne V. Shepler, University of North Texas

Kaleidoscopes
Roe Goodman's articls on The Mathematics of Mirrors and Kaleidoscopes (pdf) from the American Mathematical Monthly of April 2004 gives extensive background on the relationship between reflection groups and kaleidoscopes.

The Goodman article discusses Coxeter groups-- reflection groups in Euclidean space. However, as Shepler's definition states, reflection groups may be defined over arbitrary fields... including Galois, or finite, fields. Such fields underlie the study of Galois geometry, a part of finite geometry.

The Kaleidoscope Puzzle of Steven H. Cullinane illustrates the action of an affine reflection group in a 16-point finite geometry over the two-element Galois field, GF(2).