Coxeter group
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| Coxeter group | |
|---|---|
| Name | Coxeter group |
| Caption | Reflection symmetry in a regular polytope |
| Field | Algebra, Geometry |
| Introduced | H. S. M. Coxeter |
| Notable | Coxeter–Dynkin diagram, Weyl group, Tits representation |
Coxeter group is a class of groups generated by reflections with relations determined by pairwise orders; they appear in the study of symmetric structures, polytopes, and Lie theory. Originating in work by H. S. M. Coxeter and earlier ideas of Élie Cartan and Wilhelm Killing, Coxeter groups unify properties of reflection symmetries across Euclidean, spherical, and hyperbolic settings. These groups connect to classification results of Felix Klein, Sophus Lie, and Hermann Weyl and provide algebraic frameworks used by many mathematicians in the 20th and 21st centuries.
Definition and basic properties
A Coxeter group is defined by generators s_i with relations (s_i)^2 = 1 and (s_i s_j)^{m_{ij}} = 1 for integers m_{ij} ≥ 2 or ∞; this presentation was developed in the work of H. S. M. Coxeter and further formalized in Jacques Tits's studies. Early contributions by Wilhelm Killing and Élie Cartan influenced the identification of finite examples related to reflection symmetry in the research programs of Hermann Weyl and Évariste Galois. Fundamental algebraic properties of Coxeter groups interact with constructions studied by Issai Schur, Emmy Noether, and Emmy's contemporaries, and are central to the theory pursued by Claude Chevalley, Nathan Jacobson, and Armand Borel. Structural theorems such as the Tits alternative relate Coxeter groups to results by Jacques Tits, Jean-Pierre Serre, and André Weil.
Coxeter systems and Coxeter diagrams
A Coxeter system (W, S) pairs a Coxeter group W with its generating set S; the combinatorial data is encoded in Coxeter diagrams introduced by H. S. M. Coxeter and popularized by John Conway and Neil Sloane in their work on lattices. Diagrams closely relate to Dynkin diagrams classified by Wilhelm Killing and Élie Cartan and used by Hermann Weyl in representation theory. Coxeter diagrams are visual tools used by Egbert van Kampen, William Thurston, and Michael Atiyah in relating group presentations to topological and geometric structures. Graph-theoretic methods employed by Paul Erdős, Richard Stanley, and Gian-Carlo Rota also intersect with diagrammatic approaches.
Classification and examples
Finite Coxeter groups were classified by Coxeter and later placed in correspondence with Weyl groups studied by Élie Cartan and Hermann Weyl; the families include types A_n, B_n, D_n and exceptional types E_6, E_7, E_8, F_4, H_3, H_4, I_2(m). Classical examples connect to symmetric groups studied by Camille Jordan and Arthur Cayley, dihedral groups familiar from the work of Niels Henrik Abel, and reflection groups related to Ludwig Schläfli's polytopes. Exceptional cases appear in the work of Felix Klein on icosahedral symmetry and in applications by John McKay and Daniel Allcock. Infinite and affine Coxeter groups were developed in studies by Victor Kac, Robert Moody, and Peter Slodowy and tie to modular groups explored by Srinivasa Ramanujan and Atle Selberg.
Geometric and algebraic representations
Coxeter groups admit faithful linear representations such as the geometric representation (Tits representation) studied by Jacques Tits and later refined by Solomon Lefschetz and Jean-Pierre Serre. The groups act by isometries on Euclidean, spherical, and hyperbolic spaces, connecting to the work of Henri Poincaré, Bernhard Riemann, and Nikolai Lobachevsky. Algebraic incarnations appear as Weyl groups in Lie theory developed by Élie Cartan, Hermann Weyl, and Claude Chevalley and in Hecke algebras studied by Erich Hecke and George Lusztig. Representation-theoretic techniques used by George Mackey, Israel Gelfand, and Harish-Chandra inform understanding of character theory for Coxeter-related algebras.
Reflection groups and root systems
Reflection groups generated by orthogonal reflections were systematized by Coxeter and relate to root systems introduced by Wilhelm Killing and Élie Cartan; classical root systems A_n, B_n, C_n, D_n and exceptional roots E_6, E_7, E_8 feature prominently. Root systems underpin the classification of semisimple Lie algebras by Claude Chevalley, Nathan Jacobson, and Armand Borel and connect to lattice theory studied by John Conway and Neil Sloane. The interplay with Kac–Moody algebras was developed by Victor Kac and Robert Moody, while connections to singularity theory were explored by Vladimir Arnold and Egbert Brieskorn.
Growth, word problem, and combinatorial aspects
Growth series and the word problem for Coxeter groups involve techniques from combinatorial group theory advanced by Max Dehn, Wilhelm Magnus, and Roger Lyndon; automatic group theory contributions by David Epstein, Daniel F. Holt, and Derek F. Holt apply to many Coxeter groups. Kazhdan–Lusztig theory introduced by David Kazhdan and George Lusztig provides deep combinatorial invariants linked to Hecke algebras, with implications studied by Ian G. Macdonald, Mark Haiman, and Richard Stanley. Coxeter groups also feature in the enumerative work of Gian-Carlo Rota, Percy A. MacMahon, and Donald Knuth on reduced words, sorting networks, and Bruhat order examined by A. Borel and Jonathan Brundan.
Applications and connections to other fields
Coxeter groups appear in crystallography studied by William Barlow and Evgraf Fedorov, in the theory of regular polytopes developed by Ludwig Schläfli and H. S. M. Coxeter, and in string theory contexts pursued by Edward Witten and Michael Green. Connections to algebraic geometry involve Alexandre Grothendieck, David Mumford, and Pierre Deligne, while applications to mathematical physics draw on work by Eugene Wigner and Freeman Dyson. Computational aspects are implemented in software developed by John H. Conway's collaborators and in systems used by Richard E. Borcherds, Andrew Granville, and John Conway's circle of researchers.