Reflection group

Reflection group

A reflection group is a group action, acting on a finite dimensional vector space, which is generated by reflections: elements that fix a hyperplane in space pointwise.

For example, with regard to ordinary reflections in planes in 3D, a reflection group is an isometry group generated by these reflections. The discrete point groups in three dimensions with this property are "Cnv", "Dnh", and the together three symmetry groups of the 5 Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron). Among the frieze groups *inftyinfty and *22infty are reflection groups. Among the wallpaper groups we have pmm, p3m1, p4m, and p6m.

Each reflecting hyperplane acts as a mirror for the reflection. Reflection groups include Weyl and Coxeter groups, complex (or pseudo) reflection groups, and groups defined over arbitrary fields. Mathematical tools from geometry, topology, algebra, combinatorics, and representation theory are used to study reflection groups. For example, invariant theory (including modular), arrangements of hyperplanes, regular polytopes, Hecke algebras, Coxeter groups, Shephard groups, and braid groups all play a prominent role in investigations on reflection groups. Reflection groups also appear in coding theory, physics, chemistry, and biology.


Roe Goodman's article 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 [ a definition by Anne V. Shepler] 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.

See also

*Euclidean plane isometries as reflection group
*Coxeter group
*Weyl group
*Complex reflection group
*Regular polytope
* [ Digital Kaleidoscope] the Kaleidica

External links

* [ Reflection groups and invariant theory] , by Richard Kane (review, pdf)
*Reflection groups and Coxeter groups (ISBN 0-521-43613-3), by James E. Humphreys
* [ Jacobians of reflection groups] , by Julia Hartmann and Anne V. Shepler
* [ The Mathematics of Mirrors and Kaleidoscopes] (pdf), by Roe Goodman
* [ Reflection groups in algebraic geometry] , by Igor V. Dolgachev

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