Discrete Groups Generated by Reflections

Coxeter, H. S. M.

doi:10.2307/1968753

Cited by 395 publications

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“…2 This intimate connection with the classification of semisimple Lie groups cemented reflection groups into a central place in mathematics. The two lines of research were united by Coxeter [4] in the 1930's. Coxeter classified discrete groups generated by reflections on the n-dimensional sphere or Euclidean space.…”

Section: Some Historymentioning

confidence: 99%

“…This partial order is the subject of Chapter 2 in [1]. 4 One of the most entertaining and important sections of [1] is §4.3 on "The numbers game". Here the authors give a combinatorial method for computing generic orbits in the contragradient geometric representation and then show how to use this method to determine when a given expression s 1 · · · s n is reduced.…”

Section: Reduced Expressionsmentioning

confidence: 99%

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Book Review: Combinatorics of Coxeter groups

Davis¹

2008

Bull. Amer. Math. Soc.

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Serious work on groups generated by reflections began in the nineteenth century. In 1852 Möbius determined the finite subgroups of O(3) generated by isometric reflections on the 2-sphere (or equivalently, by orthogonal linear reflections on R 3 ). He showed that the fundamental domain for such a group on the 2-sphere was a spherical triangle with angles π p , π q , π r , with p, q, r integers ≥ 2. Since the sum of the angles in a spherical triangle is greater than π, we must have 1 p + 1 q + 1 r > 1. For p ≥ q ≥ r the only possibilities for (p, q, r) are (p, 2, 2) for any p ≥ 2 and (p, 3, 2) with p = 3, 4 or 5. The last three cases are the symmetry groups of the Platonic solids (the tetrahedron, cube or dodecahedron, respectively). Subsequent work of Riemann and Schwarz on hypergeometric functions showed the existence of groups generated by reflections across the edges of triangles with angles integral submultiples of π in either the Euclidean or hyperbolic plane. By the end of the nineteenth century, in connection with their work on automorphic forms, Klein and Poincaré had studied other groups generated by isometric reflections across the edges of polygons (with 3 or more edges) in the hyperbolic plane. 1In the second half of the nineteenth century work also began on finite reflection groups on S n for n > 2 (or equivalently, finite linear reflection groups on R n+1 ), generalizing Möbius' results for n = 2. The work developed along two lines. First, around 1850, Schläfli classified regular convex polytopes in R n+1 for n > 2. He showed that the symmetry group of such a polytope was a finite group generated by reflections and, as in Möbius' case, the projection of a fundamental domain to S n was a spherical simplex with dihedral angles integral submultiples of π. Second, around 1890, Killing and E. Cartan classified complex semisimple Lie algebras in terms of their root systems. In 1925, Weyl showed that the group of symmetries of such a root system was a finite group generated by reflections. 2 This intimate connection with the classification of semisimple Lie groups cemented reflection groups into a central place in mathematics. The two lines of research were united by Coxeter [4] in the 1930's. Coxeter classified discrete groups generated by reflections on the n-dimensional sphere or Euclidean space.The central example of a finite reflection group is the symmetric group S n+1 acting on R n+1 by permutation of coordinates. Transpositions act as orthogonal reflections across hyperplanes of R n+1 . The diagonal line L in R n+1 is fixed by S n+1 . So, S n+1 acts on the orthogonal complement L ⊥ , and L ⊥ can be identified with R n . The associated root system is type A n , and the associated complex Lie group is SL(n + 1, C). If we project the standard basis of R n+1 to R n and take its convex hull, we get a regular n-simplex. This exhibits S n+1 as the group 2 It turns out that these two cases (symmetries of regular polytopes and symmetries of root systems) cover all finite reflection groups.

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Section: Some Historymentioning

confidence: 99%

Section: Reduced Expressionsmentioning

confidence: 99%

Book Review: Combinatorics of Coxeter groups

Davis¹

2008

Bull. Amer. Math. Soc.

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“…A spherical convex polytope is nondegenerate if it does not contain opposite vertices. The classification of irreducible nondegenerate Coxeter polytopes, and hence irreducible reflection groups Γ in S n and R n with nondegenerate Γ-cell, was given by Coxeter [27]. The corresponding list of Coxeter diagrams is given in Table 1.…”

Section: By Definition Of P This Is Impossible This Proves (I)-(iii)mentioning

confidence: 99%

Reflection groups in algebraic geometry

Dolgachev

2007

Bull. Amer. Math. Soc.

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“…The theory of Coxeter groups was born from a study of finite (real) reflection groups given by H. S. M. Coxeter [Cox34,Cox35]. Although the Coxeter groups arised originally from the above geometric aspect of mathematics, Coxeter groups and their related objects (root systems, Bruhat order, Hecke algebras, etc.)…”

Section: Introductionmentioning

confidence: 99%

On the Isomorphism Problem for Coxeter Groups and Related Topics

Nuida

2014

Groups of Exceptional Type, Coxeter Groups and Related Geometries

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Discrete Groups Generated by Reflections

Cited by 395 publications

References 3 publications

Book Review: Combinatorics of Coxeter groups

Book Review: Combinatorics of Coxeter groups

Reflection groups in algebraic geometry

On the Isomorphism Problem for Coxeter Groups and Related Topics

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