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Coxeter Group

A Coxeter group is a group generated by the elements P_i for i=1, ..., n subject to

(P_iP_j)^(M_(ij))=1

for pairs with M_(ij)<infty, where M_(ij) are the elements of a Coxeter matrix. A pair with M_(ij)=infty imposes no relation. Coxeter used the notation [3^(p,q,r)] for the Coxeter group generated by the nodes of a Y-shaped Coxeter-Dynkin diagram whose three arms have p, q, and r graph edges. A Coxeter group of this form is finite iff

1/(p+1)+1/(q+1)+1/(r+1)>1.

In type B_n, signed permutations correspond to centrally symmetric noncrossing arc diagrams on 2n points (Barnard et al. 2026).

See also

Artin Group, Artin Salvetti Complex, Bimonster, Building, Coxeter-Dynkin Diagram, Coxeter Element, Coxeter Number, Coxeter Plane, Kazhdan-Lusztig Polynomial, Reduced Word

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References

Arnold, V. I. "Snake Calculus and Combinatorics of Bernoulli, Euler, and Springer Numbers for Coxeter Groups." Russian Math. Surveys 47, 3-45, 1992.Barnard, E.; Reading, N.; and Tharp, A. M. "Noncrossing Arc Diagrams of Type B." Elec. J. Combin. 33, P4.35, 2026. https://doi.org/10.37236/14125.Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension <=4." Disc. Math. 186, 69-94, 1998.Garrett, P. Buildings and Classical Groups. Boca Raton, FL: Chapman and Hall, 1997.Hsiang, W. Y. "Coxeter Groups, Weyl Reduction, and Weyl Formulas." Lec. 4 in Lectures on Lie Groups. Singapore: World Scientific, pp. 58-77, 2000.

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Coxeter Group

Cite this as:

Weisstein, Eric W. "Coxeter Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CoxeterGroup.html

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