Coxeter group

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups,^[1] and finite Coxeter groups were classified in 1935.^[2]

Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras.^[3]^[4]^[5]

^ Coxeter, H. S. M. (1934). "Discrete groups generated by reflections". Annals of Mathematics. 35 (3): 588–621. CiteSeerX 10.1.1.128.471. doi:10.2307/1968753. JSTOR 1968753.
^ Coxeter, H. S. M. (January 1935). "The complete enumeration of finite groups of the form $r_{i}^{2}=(r_{i}r_{j})^{k_{ij}}=1$ ". Journal of the London Mathematical Society: 21–25. doi:10.1112/jlms/s1-10.37.21.
^ Bourbaki, Nicolas (2002). "4-6". Lie Groups and Lie Algebras. Elements of Mathematics. Springer. ISBN 978-3-540-42650-9. Zbl 0983.17001.
^ Humphreys, James E. (1990). Reflection Groups and Coxeter Groups (PDF). Cambridge Studies in Advanced Mathematics. Vol. 29. Cambridge University Press. doi:10.1017/CBO9780511623646. ISBN 978-0-521-43613-7. Zbl 0725.20028. Retrieved 2023-11-18.
^ Davis, Michael W. (2007). The Geometry and Topology of Coxeter Groups (PDF). Princeton University Press. ISBN 978-0-691-13138-2. Zbl 1142.20020. Retrieved 2023-11-18.