In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... H.S.M. Coxeter. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, one method of defining a group is by a presentation. ... The symmetry groups of the 5 Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) are generated by reflections and rotations in space. ... The symmetry group of an object (e. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ...

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. The symmetry group of an object (e. ... A dodecahedron, one of the five Platonic solids. ... In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes orthogonal to the roots. ... In mathematics, a simple Lie group is a Lie group which is also a simple group. ... In mathematics, the triangle groups are groups that can be realized geometrically by sequences of reflections across the sides of certain triangles. ... A tessellated plane seen in street pavement. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... In mathematics, a Kac–Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system. ...

Definition

Formally, a Coxeter group can be defined as a group with the presentation This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, one method of defining a group is by a presentation. ...

where m_ii = 1 and m_ij ≥ 2 for i ≠ j. The condition m_ij = ∞ means no relation of the form (r_i r_j)^m should be imposed.

A number of conclusions can be drawn immediately from the above definition.

• The relation m_ii = 1 means that (r_i)² = 1 for all i ; the generators are involutions.
• If m_ij = 2, then the generators r_i and r_j commute. This follows by observing that

xx = yy = 1,

together with

xyxy = 1

implies that

xy = xxyxyy = yx.

• In order to avoid redundancy among the relations, it is necessary to assume that m_ij=m_ji. This follows by observing that

yy = 1,

together with

(xy)^m = 1

implies that

(yx)^m = (yx)^myy = y(xy)^my = yy = 1.

The Coxeter matrix is the n×n, symmetric matrix with entries m_ij. Indeed, every symmetric matrix with positive integer and ∞ entries and with 1's on the diagonal serves to define a Coxeter group. The Coxeter matrix can be conveniently encoded by a Coxeter graph, as per the following rules. In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...

• The vertices of the graph are labelled by generator subscripts.
• Vertices i and j are connected if and only if m_ij ≥ 3.
• An edge is labelled with the value of m_ij whenever it is 4 or greater.

In particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter graph has two or more connected components, the associated group is the direct product of the groups associated to the individual components. Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ... In an undirected graph, a connected component or component is a maximal connected subgraph. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ...

An example

The graph in which vertices 1 through n are placed in a row with each vertex connected by an unlabelled edge to its immediate neighbors gives rise to the symmetric group S_n+1; the generators correspond to the transpositions (1 2), (2 3), ... (n n+1). Two non-consecutive transpositions always commute, while (k k+1) (k+1 k+2) gives the 3-cycle (k k+1 k+2). Of course this only shows that S_n+1 is a quotient group of the Coxeter group, but it is not too difficult to check that equality holds. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...

Finite Coxeter groups

Coxeter graphs of the finite Coxeter groups.

Every Weyl group can be realized as a Coxeter group. The Coxeter graph can be obtained from the Dynkin diagram by replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. The example given above corresponds to the Weyl group of the root system of type A_n. The Weyl groups include most of the finite Coxeter groups, but there are additional examples as well. The following list gives all connected Coxeter graphs giving rise to finite groups: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes orthogonal to the roots. ... See also Simple Lie group. ... In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ... In mathematics and computer science, graph theory studies the properties of graphs. ...

Comparing this with the list of simple root systems, we see that B_n and C_n give rise to the same Coxeter group. Also, G₂ appears to be missing, but it is present under the name I₂(6). The additions to the list are H₃, H₄, and the I₂(p).

Some properties of the finite Coxeter groups are given in the following table:

In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... Coxeter groups in the plane with equivalent diagrams. ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... Image File history File links CDW_dot. ... Image File history File links CDW_3b. ... Image File history File links CDW_dot. ... Image File history File links CDW_3b. ... Image File history File links CDW_3b. ... Image File history File links CDW_dot. ... Image File history File links CDW_3b. ... Image File history File links CDW_dot. ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3b. ... Image File history File links CDW_3b. ... Image File history File links CDW_dot. ... Image File history File links CDW_3b. ... Image File history File links CDW_dot. ... In geometry, demihypercubes (also called half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-00. ... Image File history File links CD_3b. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... Image File history File links CDW_dot. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3b. ... Image File history File links CDW_dot. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... Image File history File links CDW_dot. ... Image File history File links CDW_3b. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3b. ... Image File history File links CDW_dot. ... In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is sometimes thought of as the 4-dimensional analog of the dodecahedron. ... Vertex figure: icosahedron In geometry, the 600-cell (or hexacosichoron) is the convex regular 4-polytope with Schläfli symbol {3,3,5}. It is sometimes thought of as the 4-dimensional analog of the icosahedron. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3b. ... Image File history File links CDW_dot. ... Image File history File links CDW_3b. ... Image File history File links CDW_dot. ... In mathematics, the E6 polytope is the convex hull of the roots of E6. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-00. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... In mathematics, E7 is the name of several Lie groups and also their Lie algebras . ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-00. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... In mathematics, E8 is the name given to a family of closely related structures. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-00. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_dot. ...

Symmetry groups of regular polytopes

All symmetry groups of regular polytopes are finite Coxeter groups. The dihedral groups, which are the symmetry groups of regular polygons, form the series I₂(p). The symmetry group of a regular n-simplex is the symmetric group S_n+1, also known as the Coxeter group of type A_n. The symmetry group of the n-cube is the same as that of the n-cross-polytope, namely BC_n. The symmetry group of the regular dodecahedron and the regular icosahedron is H₃. In dimension 4, there are three special regular polytopes, the 24-cell, the 120-cell, and the 600-cell. The first has symmetry group F₄, while the other two have symmetry group H₄. The symmetry group of an object (e. ... A dodecahedron, one of the five Platonic solids. ... This article may be confusing for some readers, and should be edited to enhance clarity. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is sometimes thought of as the 4-dimensional analog of the dodecahedron. ... Vertex figure: icosahedron In geometry, the 600-cell (or hexacosichoron) is the convex regular 4-polytope with Schläfli symbol {3,3,5}. It is sometimes thought of as the 4-dimensional analog of the icosahedron. ...

The Coxeter groups of type D_n, E₆, E₇, and E₈ are the symmetry groups of certain semiregular polytopes.

Affine Weyl groups

The affine Weyl groups form a second important series of Coxeter groups. These are not finite themselves, but each contains a normal abelian subgroup such that the corresponding quotient group is finite. In each case, the quotient group is itself a Weyl group, and the Coxeter graph is obtained from the Coxeter graph of the Weyl group by adding an additional vertex and one or two additional edges. For example, for n ≥ 2, the graph consisting of n+1 vertices in a circle is obtained from A_n in this way, and the corresponding Coxeter group is the affine Weyl group of A_n. For n = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles. In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...

A list of the Affine Coxeter groups follows:

Image File history File links No higher resolution available. ...

Note the subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's graph.

Hyperbolic Coxeter groups

There are also hyperbolic Coxeter groups describing reflection groups in hyperbolic geometry. Lines through a given point P and hyperparallel to line l. ...

Bruhat order

Choice of reflection generators gives rise to a length function l on a Coxeter group, namely the minimum number of uses of generators required to express a group element. From that the Bruhat order, a partial order relation, is defined: an element v exceeds an element u if (one step) it has length which is one greater and is the product of u with a reflection generator, or (any number) it exceeds u in the transitive closure of the one-step relation. In other words, u ≤ v means that v is built up from u with the appropriate number l(v) − l(u) of generating reflections. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, the transitive closure of a binary relation R on a set X is the smallest transitive relation on X that contains R. For any relation R the transitive closure of R always exists. ...

References

• Larry C Grove and Clark T. Benson, Finite Reflection Groups, Graduate texts in mathematics, vol. 99, Springer, (1985)
• James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990)
• Richard Kane, Reflection Groups and Invariant Theory, CMS Books in Mathematics, Springer (2001)
• Anders Björner and Francesco Brenti, Combinatorics of Coxeter Groups, Graduate texts in mathematics, vol. 231, Springer, (2005)

See also

• Artin group
• Weyl group
• Triangle group
• Coxeter number
• Complex reflection group
• Kazhdan-Lusztig polynomial

In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form where . For , represents an alternating product of and of length , beginning with . ... In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes orthogonal to the roots. ... In mathematics, the triangle groups are groups that can be realized geometrically by sequences of reflections across the sides of certain triangles. ... The introduction of this article does not provide enough context for readers unfamiliar with the subject. ... In mathematics, a complex reflection group is a group acting on a finite-dimensional complex vector space, that is generated by complex reflections: elements that fix a complex hyperplane in space pointwise. ... In representation theory, a Kazhdan–Lusztig polynomial is a member of a family of integral polynomials introduced in work of David Kazhdan and George Lusztig (Kazhdan & Lusztig 1979). ...

External links

• Eric W. Weisstein, Coxeter group at MathWorld.
• Jenn, software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators.