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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, and finite Coxeter groups were classified in 1935.

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.

Definition

Formally, a Coxeter group can be defined as a group with the presentation

${\displaystyle \left\langle r_{1},r_{2},\ldots ,r_{n}\mid (r_{i}r_{j})^{m_{ij}}=1\right\rangle }$

where ${\displaystyle m_{ii}=1}$ and ${\displaystyle m_{ij}=m_{ji}\geq 2}$ is either an integer or ${\displaystyle \infty }$ for ${\displaystyle i\neq j}$. Here, the condition ${\displaystyle m_{ij}=\infty }$ means that no relation of the form ${\displaystyle (r_{i}r_{j})^{m}=1}$ for any integer ${\displaystyle m\geq 2}$ should be imposed.

The pair ${\displaystyle (W,S)}$ where ${\displaystyle W}$ is a Coxeter group with generators ${\displaystyle S=\{r_{1},\dots ,r_{n}\}}$ is called a Coxeter system. Note that in general ${\displaystyle S}$ is not uniquely determined by ${\displaystyle W}$. For example, the Coxeter groups of type ${\displaystyle B_{3}}$ and ${\displaystyle A_{1}\times A_{3}}$ are isomorphic but the Coxeter systems are not equivalent, since the former has 3 generators and the latter has 1 + 3 = 4 generators (see below for an explanation of this notation).

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

• The relation ${\displaystyle m_{ii}=1}$ means that ${\displaystyle (r_{i}r_{i})^{1}=(r_{i})^{2}=1}$ for all ${\displaystyle i}$ ; as such the generators are involutions.
• If ${\displaystyle m_{ij}=2}$, then the generators ${\displaystyle r_{i}}$ and ${\displaystyle r_{j}}$ commute. This follows by observing that

${\displaystyle xx=yy=1}$,

together with

${\displaystyle xyxy=1}$

implies that

${\displaystyle xy=x(xyxy)y=(xx)yx(yy)=yx}$.

Alternatively, since the generators are involutions, ${\displaystyle r_{i}=r_{i}^{-1}}$, so ${\displaystyle 1=(r_{i}r_{j})^{2}=r_{i}r_{j}r_{i}r_{j}=r_{i}r_{j}r_{i}^{-1}r_{j}^{-1}}$. That is to say, the commutator of ${\displaystyle r_{i}}$ and ${\displaystyle r_{j}}$ is equal to 1, or equivalently that ${\displaystyle r_{i}}$ and ${\displaystyle r_{j}}$ commute.

The reason that ${\displaystyle m_{ij}=m_{ji}}$ for ${\displaystyle i\neq j}$ is stipulated in the definition is that

${\displaystyle yy=1}$,

together with

${\displaystyle (xy)^{m}=1}$

already implies that

${\displaystyle (yx)^{m}=(yx)^{m}yy=y(xy)^{m}y=yy=1}$.

An alternative proof of this implication is the observation that ${\displaystyle (xy)^{k}}$ and ${\displaystyle (yx)^{k}}$ are conjugates: indeed ${\displaystyle y(xy)^{k}y^{-1}=(yx)^{k}yy^{-1}=(yx)^{k}}$.

Coxeter matrix and Schläfli matrix

The Coxeter matrix is the ${\displaystyle n\times n}$ symmetric matrix with entries ${\displaystyle m_{ij}}$. Indeed, every symmetric matrix with diagonal entries exclusively 1 and nondiagonal entries in the set ${\displaystyle \{2,3,\ldots \}\cup \{\infty \}}$ is a Coxeter matrix.

The Coxeter matrix can be conveniently encoded by a Coxeter diagram, as per the following rules.

• The vertices of the graph are labelled by generator subscripts.
• Vertices ${\displaystyle i}$ and ${\displaystyle j}$ are adjacent if and only if ${\displaystyle m_{ij}\geq 3}$.
• An edge is labelled with the value of ${\displaystyle m_{ij}}$ whenever the value is ${\displaystyle 4}$ or greater.

In particular, two generators commute if and only if they are not joined 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. Thus the disjoint union of Coxeter graphs yields a direct product of Coxeter groups.

The Coxeter matrix, ${\displaystyle M_{ij}}$, is related to the ${\displaystyle n\times n}$ Schläfli matrix ${\displaystyle C}$ with entries ${\displaystyle C_{ij}=-2\cos(\pi /M_{ij})}$, but the elements are modified, being proportional to the dot product of the pairwise generators. The Schläfli matrix is useful because its eigenvalues determine whether the Coxeter group is of finite type (all positive), affine type (all non-negative, at least one zero), or indefinite type (otherwise). The indefinite type is sometimes further subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups.

Examples

Coxeter group	A1×A1	A2	B2	I2(5)	G2	${\displaystyle {\tilde {A}}_{1}=I_{2}(\infty )}$	A3	B3	D4	${\displaystyle {\tilde {A}}_{3}}$
Coxeter diagram
Coxeter matrix	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$
Schläfli matrix	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$

An example

The graph ${\displaystyle A_{n}}$ in which vertices ${\displaystyle 1}$ through ${\displaystyle n}$ are placed in a row with each vertex joined by an unlabelled edge to its immediate neighbors is the Coxeter diagram of the symmetric group ${\displaystyle S_{n+1}}$; the generators correspond to the transpositions ${\displaystyle (1~~2),(2~~3),\dots ,(n~~n+1)}$. Any two non-consecutive transpositions commute, while multiplying two consecutive transpositions gives a 3-cycle : ${\displaystyle (k~~k+1)\cdot (k+1~~k+2)=(k~~k+2~~k+1)}$. Therefore ${\displaystyle S_{n+1}}$ is a quotient of the Coxeter group having Coxeter diagram ${\displaystyle A_{n}}$. Further arguments show that this quotient map is an isomorphism.

Abstraction of reflection groups

Coxeter groups are an abstraction of reflection groups. Coxeter groups are abstract groups, in the sense of being given via a presentation. On the other hand, reflection groups are concrete, in the sense that each of its elements is the composite of finitely many geometric reflections about linear hyperplanes in some euclidean space. Technically, a reflection group is a subgroup of a linear group (or various generalizations) generated by orthogonal matrices of determinant -1. Each generator of a Coxeter group has order 2, which abstracts the geometric fact that performing a reflection twice is the identity. Each relation of the form ${\displaystyle (r_{i}r_{j})^{k}}$, corresponding to the geometric fact that, given two hyperplanes meeting at an angle of ${\displaystyle \pi /k}$, the composite of the two reflections about these hyperplanes is a rotation by ${\displaystyle 2\pi /k}$, which has order k.

In this way, every reflection group may be presented as a Coxeter group. The converse is partially true: every finite Coxeter group admits a faithful representation as a finite reflection group of some Euclidean space. However, not every infinite Coxeter group admits a representation as a reflection group.

Finite Coxeter groups have been classified.

Finite Coxeter groups

Classification

Finite Coxeter groups are classified in terms of their Coxeter diagrams.

The finite Coxeter groups with connected Coxeter diagrams consist of three one-parameter families of increasing dimension (${\displaystyle A_{n}}$ for ${\displaystyle n\geq 1}$, ${\displaystyle B_{n}}$ for ${\displaystyle n\geq 2}$, and ${\displaystyle D_{n}}$ for ${\displaystyle n\geq 4}$), a one-parameter family of dimension two (${\displaystyle I_{2}(p)}$ for ${\displaystyle p\geq 5}$), and six exceptional groups (${\displaystyle E_{6},E_{7},E_{8},F_{4},H_{3},}$ and ${\displaystyle H_{4}}$). Every finite Coxeter group is the direct product of finitely many of these irreducible groups.

Weyl groups

Many, but not all of these, are Weyl groups, and every Weyl group can be realized as a Coxeter group. The Weyl groups are the families ${\displaystyle A_{n},B_{n},}$ and ${\displaystyle D_{n},}$ and the exceptions ${\displaystyle E_{6},E_{7},E_{8},F_{4},}$ and ${\displaystyle I_{2}(6),}$ denoted in Weyl group notation as ${\displaystyle G_{2}.}$

The non-Weyl ones are the exceptions ${\displaystyle H_{3}}$ and ${\displaystyle H_{4},}$ and those members of the family ${\displaystyle I_{2}(p)}$ that are not exceptionally isomorphic to a Weyl group (namely ${\displaystyle I_{2}(3)\cong A_{2},I_{2}(4)\cong B_{2},}$ and ${\displaystyle I_{2}(6)\cong G_{2}}$).

This can be proven by comparing the restrictions on (undirected) Dynkin diagrams with the restrictions on Coxeter diagrams of finite groups: formally, the Coxeter graph can be obtained from the Dynkin diagram by discarding the direction of the edges, and replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. Also note that every finitely generated Coxeter group is an automatic group. Dynkin diagrams have the additional restriction that the only permitted edge labels are 2, 3, 4, and 6, which yields the above. Geometrically, this corresponds to the crystallographic restriction theorem, and the fact that excluded polytopes do not fill space or tile the plane – for ${\displaystyle H_{3},}$ the dodecahedron (dually, icosahedron) does not fill space; for ${\displaystyle H_{4},}$ the 120-cell (dually, 600-cell) does not fill space; for ${\displaystyle I_{2}(p)}$ a p-gon does not tile the plane except for ${\displaystyle p=3,4,}$ or ${\displaystyle 6}$ (the triangular, square, and hexagonal tilings, respectively).

Note further that the (directed) Dynkin diagrams B_n and C_n give rise to the same Weyl group (hence Coxeter group), because they differ as directed graphs, but agree as undirected graphs – direction matters for root systems but not for the Weyl group; this corresponds to the hypercube and cross-polytope being different regular polytopes but having the same symmetry group.

Properties

Some properties of the finite irreducible Coxeter groups are given in the following table. The order of a reducible group can be computed by the product of its irreducible subgroup orders.

Rank n	Group symbol	Alternate symbol	Bracket notation	Coxeter graph	Reflections m = 1⁄2nh	Coxeter number h	Order	Group structure	Related polytopes
1	A1	A1			1	2	2	${\displaystyle S_{2}}$	{ }
2	A2	A2			3	3	6	${\displaystyle S_{3}\cong D_{6}\cong \operatorname {GO} _{2}^{-}(2)\cong \operatorname {GO} _{2}^{+}(4)}$	{3}
3	A3	A3			6	4	24	${\displaystyle S_{4}}$	{3,3}
4	A4	A4			10	5	120	${\displaystyle S_{5}}$	{3,3,3}
5	A5	A5			15	6	720	${\displaystyle S_{6}}$	{3,3,3,3}
n	An	An		...	n(n + 1)/2	n + 1	(n + 1)!	${\displaystyle S_{n+1}}$	n-simplex
2	B2	C2			4	4	8	${\displaystyle C_{2}\wr S_{2}\cong D_{8}\cong \operatorname {GO} _{2}^{-}(3)\cong \operatorname {GO} _{2}^{+}(5)}$	{4}
3	B3	C3			9	6	48	${\displaystyle C_{2}\wr S_{3}\cong S_{4}\times 2}$	{4,3} / {3,4}
4	B4	C4			16	8	384	${\displaystyle C_{2}\wr S_{4}}$	{4,3,3} / {3,3,4}
5	B5	C5			25	10	3840	${\displaystyle C_{2}\wr S_{5}}$	{4,3,3,3} / {3,3,3,4}
n	Bn	Cn		...	n	2n	2 n!	${\displaystyle C_{2}\wr S_{n}}$	n-cube / n-orthoplex
4	D4	B4			12	6	192	${\displaystyle C_{2}^{3}S_{4}\cong 2^{1+4}\colon S_{3}}$	h{4,3,3} / {3,3}
5	D5	B5			20	8	1920	${\displaystyle C_{2}^{4}S_{5}}$	h{4,3,3,3} / {3,3,3}
n	Dn	Bn		...	n(n − 1)	2(n − 1)	2 n!	${\displaystyle C_{2}^{n-1}S_{n}}$	n-demicube / n-orthoplex
6	E6	E6			36	12	51840 (72x6!)	${\displaystyle \operatorname {GO} _{6}^{-}(2)\cong \operatorname {SO} _{5}(3)\cong \operatorname {PSp} _{4}(3)\colon 2\cong \operatorname {PSU} _{4}(2)\colon 2}$	221, 122
7	E7	E7			63	18	2903040 (72x8!)	${\displaystyle \operatorname {GO} _{7}(2)\times 2\cong \operatorname {Sp} _{6}(2)\times 2}$	321, 231, 132
8	E8	E8			120	30	696729600 (192x10!)	${\displaystyle 2\cdot \operatorname {GO} _{8}^{+}(2)}$	421, 241, 142
4	F4	F4			24	12	1152	${\displaystyle \operatorname {GO} _{4}^{+}(3)\cong 2^{1+4}\colon (S_{3}\times S_{3})}$	{3,4,3}
2	G2	– (D 2)			6	6	12	${\displaystyle D_{12}\cong \operatorname {GO} _{2}^{-}(5)\cong \operatorname {GO} _{2}^{+}(7)}$	{6}
2	I2(5)	G2			5	5	10	${\displaystyle D_{10}\cong \operatorname {GO} _{2}^{-}(4)}$	{5}
3	H3	G3			15	10	120	${\displaystyle 2\times A_{5}}$	{3,5} / {5,3}
4	H4	G4			60	30	14400	${\displaystyle 2\cdot (A_{5}\times A_{5})\colon 2}$	{5,3,3} / {3,3,5}
2	I2(n)	D 2			n	n	2n	${\displaystyle D_{2n}}$ ${\displaystyle \cong \operatorname {GO} _{2}^{-}(n-1)}$ when n = p + 1, p prime ${\displaystyle \cong \operatorname {GO} _{2}^{+}(n+1)}$ when n = p − 1, p prime	{p}

Symmetry groups of regular polytopes

The symmetry group of every regular polytope is a finite Coxeter group. Note that dual polytopes have the same symmetry group.

There are three series of regular polytopes in all dimensions. 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 and its dual, the n-cross-polytope, is B_n, and is known as the hyperoctahedral group.

The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter groups. In two dimensions, the dihedral groups, which are the symmetry groups of regular polygons, form the series I₂(p), for p ≥ 3. In three dimensions, the symmetry group of the regular dodecahedron and its dual, the regular icosahedron, is H₃, known as the full icosahedral group. In four dimensions, there are three exceptional regular polytopes, the 24-cell, the 120-cell, and the 600-cell. The first has symmetry group F₄, while the other two are dual and have symmetry group H₄.

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

A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples.

Polytope elements

Polygon (2-polytope)

• Vertex the ridge or (n−2)-face of the polygon
• Edge the facet or (n−1)-face of the polygon

Polyhedron (3-polytope)

• Vertex the peak or (n−3)-face of the polyhedron
• Edge the ridge or (n−2)-face of the polyhedron
• Face the facet or (n−1)-face of the polyhedron

Polychoron (4-polytope)

• Vertex the (n−4)-face of the polychoron
• Edge the peak or (n−3)-face of the polychoron
• Face the ridge or (n−2)-face of the polychoron
• Cell the facet or (n−1)-face of the polychoron

5-polytope

• Vertex the (n−5)-face of the 5-polytope
• Edge the (n−4)-face of the 5-polytope
• Face the peak or (n−3)-face of the 5-polytope
• Cell the ridge or (n−2)-face of the 5-polytope
• Hypercell or Teron the facet or (n−1)-face of the 5-polytope

Other

• Point
• Line segment
• Vertex figure
• Peak – (n−3)-face
• Ridge – (n−2)-face
• Facet – (n−1)-face

Two dimensional (polygons)

• Triangle
  • Equilateral triangle
  • Isosceles triangle
    • Golden triangle (mathematics)
  • Scalene triangle
  • Right triangle
  • Oblique triangle
    • Acute triangle
    • Obtuse Triangle

• Quadrilateral
  • Rectangle
    • Square
      • Unit square
    • Oblong
      • Golden rectangle
      • Silver rectangle
      • Ailles rectangle
  • Rhombus
    • Golden Rhombus
  • Parallelogram
    • Rhomboid
  • Trapezoid
    • Isosceles trapezoid
  • Kite
    • Lute of pythagoras
    • Right Kite
  • Antiparallelogram

• Pentagon
• Hexagon
• Heptagon
• Octagon
• Nonagon
• Decagon
• Hendecagon
• Dodecagon
• Triskaidecagon
• Tetradecagon
• Pentadecagon
• Hexadecagon
• Heptadecagon
• Octadecagon
• Enneadecagon
• Icosagon
• Icosihenagon
• Icosidigon
• Icositrigon
• Icositetragon
• Icosipentagon
• Icosihexagon
• Icosiheptagon
• Icosioctagon
• Icosienneagon
• Triacontagon
• Tetracontagon
• Pentacontagon
• Hexacontagon
• Heptacontagon
• Octacontagon
• Enneacontagon
• Hectogon
• 257-gon
• Chiliagon
• Myriagon
• 65537-gon
• Megagon
• Gigagon
• Teragon
• Apeirogon

Star polygons

• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
• Icositetragram

Families

• Concave polygon
• Cyclic polygon
• Regular polygon
• Polyform
• Gnomon
• Golygon

Tilings

List of uniform tilings

Uniform tilings in hyperbolic plane

Archimedean tiling

• Square tiling
• Triangular tiling
• Hexagonal tiling
• Truncated square tiling
• Snub square tiling
• Trihexagonal tiling
• Truncated hexagonal tiling
• Rhombitrihexagonal tiling
• Truncated trihexagonal tiling
• Snub hexagonal tiling
• Elongated triangular tiling

Three dimensional (polyhedra)

Three-dimensional space

Regular

Regular polyhedron

• Platonic solid:
  • Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron
• Regular spherical polyhedron
  • Dihedron, Hosohedron
• Kepler–Poinsot polyhedron (Regular star polyhedra)
  • Small stellated dodecahedron, Great stellated dodecahedron, Great icosahedron, Great dodecahedron
• Abstract regular polyhedra (Projective polyhedron)
  • Hemicube (geometry), hemi-octahedron, hemi-dodecahedron, hemi-icosahedron

Tetrahedron

• Disphenoid

Pentahedron

• Square pyramid, Triangular prism

Hexahedron

• Parallelepiped, Cuboid, Rhombohedron, Trigonal trapezohedron, Cube, Pentagonal pyramid, Triangular bipyramid, quadrilateral frustum

Heptahedron

• hexagonal pyramid, pentagonal prism, tetrahemihexahedron

Octahedron

• Hexagonal prism, Truncated tetrahedron, Tetragonal trapezohedron

Enneahedron

• Octagonal pyramid, Heptagonal prism

Decahedron

• Octagonal prism, Square antiprism, Square cupola, Pentagonal bipyramid, Augmented pentagonal prism

Dodecahedron

• Pentagonal antiprism, Decagonal prism, Pentagonal cupola, Snub disphenoid, Elongated square bipyramid, Metabidiminished icosahedron, Hexagonal bipyramid, Hexagonal trapezohedron, Triakis tetrahedron, Rhombic dodecahedron, Hendecagonal pyramid, Trapezo-rhombic dodecahedron, Rhombo-hexagonal dodecahedron

Archimedean solids

Archimedean solid

• Truncated tetrahedron, Cuboctahedron, Truncated cube, Truncated octahedron, Rhombicuboctahedron, Truncated cuboctahedron, Snub cube, Icosidodecahedron, Truncated dodecahedron, Truncated icosahedron, Rhombicosidodecahedron, Truncated icosidodecahedron, Snub dodecahedron

Prisms and antiprisms

Prism

• Triangular prism, Pentagonal prism, Hexagonal prism, Heptagonal prism, Octagonal prism, Enneagonal prism, Decagonal prism, Hendecagonal prism, Dodecagonal prism

Antiprism

• Square antiprism, Pentagonal antiprism, Hexagonal antiprism, Heptagonal antiprism, Octagonal antiprism, Enneagonal antiprism, Decagonal antiprism, Dodecagonal antiprism

Catalan solids

Catalan solid

• Triakis tetrahedron, Rhombic dodecahedron, Triakis octahedron, Tetrakis hexahedron, Deltoidal icositetrahedron, Disdyakis dodecahedron, Pentagonal icositetrahedron, Rhombic triacontahedron, Triakis icosahedron, Pentakis dodecahedron, Deltoidal hexecontahedron, Disdyakis triacontahedron, Pentagonal hexecontahedron

Bipyramids and Trapezohedron

• Bipyramid
  • Triangular bipyramid, Pentagonal bipyramid, Hexagonal bipyramid, Heptagonal bipyramid, Octagonal bipyramid, Decagonal bipyramid
• Trapezohedron

Uniform star polyhedra

Uniform star polyhedron

• Cubitruncated cuboctahedron
• Cubohemioctahedron
• Ditrigonal dodecadodecahedron
• Dodecadodecahedron
• Great cubicuboctahedron
• Great dirhombicosidodecahedron
• Great disnub dirhombidodecahedron
• Great ditrigonal dodecicosidodecahedron
• Great ditrigonal icosidodecahedron
• Great dodecahemicosahedron
• Great dodecahemidodecahedron
• Great dodecicosahedron
• Great dodecicosidodecahedron
• Great icosicosidodecahedron
• Great icosidodecahedron
• Great icosihemidodecahedron
• Great inverted snub icosidodecahedron
• Great retrosnub icosidodecahedron
• Great rhombidodecahedron
• Great rhombihexahedron
• Great snub dodecicosidodecahedron
• Great snub icosidodecahedron
• Great stellated truncated dodecahedron
• Great truncated cuboctahedron
• Great truncated icosidodecahedron
• Icosidodecadodecahedron
• Icositruncated dodecadodecahedron
• Inverted snub dodecadodecahedron
• Nonconvex great rhombicosidodecahedron
• Nonconvex great rhombicuboctahedron
• Octahemioctahedron
• Rhombicosahedron
• Rhombidodecadodecahedron
• Small cubicuboctahedron
• Small ditrigonal dodecicosidodecahedron
• Small ditrigonal icosidodecahedron
• Small dodecahemicosahedron
• Small dodecahemidodecahedron
• Small dodecicosahedron
• Small dodecicosidodecahedron
• Small icosicosidodecahedron
• Small icosihemidodecahedron
• Small retrosnub icosicosidodecahedron
• Small rhombidodecahedron
• Small rhombihexahedron
• Small snub icosicosidodecahedron
• Small stellated truncated dodecahedron
• Snub dodecadodecahedron
• Snub icosidodecadodecahedron
• Stellated truncated hexahedron
• Tetrahemihexahedron
• Truncated dodecadodecahedron
• Truncated great dodecahedron
• Truncated great icosahedron

Uniform prismatic star polyhedra

Prismatic uniform polyhedron

• Pentagrammic prism, Pentagrammic antiprism, Pentagrammic crossed-antiprism
• Heptagrammic antiprism (7/2), Heptagrammic antiprism (7/3)
• Enneagrammic antiprism (9/2). Enneagrammic antiprism (9/4)
• Enneagrammic crossed-antiprism, Enneagrammic prism (9/2), Enneagrammic prism (9/4)
• Decagrammic prism, Decagrammic antiprism

Johnson solids

Johnson solid

1. Augmented dodecahedron
2. Augmented hexagonal prism
3. Augmented pentagonal prism
4. Augmented sphenocorona
5. Augmented triangular prism
6. Augmented tridiminished icosahedron
7. Augmented truncated cube
8. Augmented truncated dodecahedron
9. Augmented truncated tetrahedron
10. Biaugmented pentagonal prism
11. Biaugmented triangular prism
12. Biaugmented truncated cube
13. Bigyrate diminished rhombicosidodecahedron
14. Bilunabirotunda
15. Diminished rhombicosidodecahedron
16. Disphenocingulum
17. Elongated pentagonal bipyramid
18. Elongated pentagonal cupola
19. Elongated pentagonal gyrobicupola
20. Elongated pentagonal gyrobirotunda
21. Elongated pentagonal gyrocupolarotunda
22. Elongated pentagonal orthobicupola
23. Elongated pentagonal orthobirotunda
24. Elongated pentagonal orthocupolarotunda
25. Elongated pentagonal pyramid
26. Elongated pentagonal rotunda
27. Elongated square bipyramid
28. Elongated square cupola
29. Elongated square gyrobicupola
30. Elongated square pyramid
31. Elongated triangular bipyramid
32. Elongated triangular cupola
33. Elongated triangular gyrobicupola
34. Elongated triangular orthobicupola
35. Elongated triangular pyramid
36. Gyrate bidiminished rhombicosidodecahedron
37. Gyrate rhombicosidodecahedron
38. Gyrobifastigium
39. Gyroelongated pentagonal bicupola
40. Gyroelongated pentagonal birotunda
41. Gyroelongated pentagonal cupola
42. Gyroelongated pentagonal cupolarotunda
43. Gyroelongated pentagonal pyramid
44. Gyroelongated pentagonal rotunda
45. Gyroelongated square bicupola
46. Gyroelongated square bipyramid
47. Gyroelongated square cupola
48. Gyroelongated square pyramid
49. Gyroelongated triangular bicupola
50. Gyroelongated triangular cupola
51. Hebesphenomegacorona
52. Metabiaugmented dodecahedron
53. Metabiaugmented hexagonal prism
54. Metabiaugmented truncated dodecahedron
55. Metabidiminished icosahedron
56. Metabidiminished rhombicosidodecahedron
57. Metabigyrate rhombicosidodecahedron
58. Metagyrate diminished rhombicosidodecahedron
59. Parabiaugmented dodecahedron
60. Parabiaugmented hexagonal prism
61. Parabiaugmented truncated dodecahedron
62. Parabidiminished rhombicosidodecahedron
63. Parabigyrate rhombicosidodecahedron
64. Paragyrate diminished rhombicosidodecahedron
65. Pentagonal bipyramid
66. Pentagonal cupola
67. Pentagonal gyrobicupola
68. Pentagonal gyrocupolarotunda
69. Pentagonal orthobicupola
70. Pentagonal orthobirotunda
71. Pentagonal orthocupolarotunda
72. Pentagonal pyramid
73. Pentagonal rotunda
74. Snub disphenoid
75. Snub square antiprism
76. Sphenocorona
77. Sphenomegacorona
78. Square cupola
79. Square gyrobicupola
80. Square orthobicupola
81. Square pyramid
82. Triangular bipyramid
83. Triangular cupola
84. Triangular hebesphenorotunda
85. Triangular orthobicupola
86. Triaugmented dodecahedron
87. Triaugmented hexagonal prism
88. Triaugmented triangular prism
89. Triaugmented truncated dodecahedron
90. Tridiminished icosahedron
91. Tridiminished rhombicosidodecahedron
92. Trigyrate rhombicosidodecahedron

Dual uniform star polyhedra

• Great complex icosidodecahedron
• Great deltoidal hexecontahedron
• Great deltoidal icositetrahedron
• Great dirhombicosidodecacron
• Great dirhombicosidodecahedron
• Great disdyakis dodecahedron
• Great disdyakis triacontahedron
• Great disnub dirhombidodecacron
• Great ditrigonal dodecacronic hexecontahedron
• Great dodecacronic hexecontahedron
• Great dodecahemicosacron
• Great dodecicosacron
• Great hexacronic icositetrahedron
• Great hexagonal hexecontahedron
• Great icosacronic hexecontahedron
• Great icosihemidodecacron
• Great inverted pentagonal hexecontahedron
• Great pentagonal hexecontahedron
• Great pentagrammic hexecontahedron
• Great pentakis dodecahedron
• Great rhombic triacontahedron
• Great rhombidodecacron
• Great rhombihexacron
• Great stellapentakis dodecahedron
• Great triakis icosahedron
• Great triakis octahedron
• Great triambic icosahedron
• Medial deltoidal hexecontahedron
• Medial disdyakis triacontahedron
• Medial hexagonal hexecontahedron
• Medial icosacronic hexecontahedron
• Medial inverted pentagonal hexecontahedron
• Medial pentagonal hexecontahedron
• Medial rhombic triacontahedron
• Hexahemioctacron
• Hemipolyhedron
• Octahemioctacron
• Rhombicosacron
• Small complex icosidodecahedron
• Small ditrigonal dodecacronic hexecontahedron
• Small dodecacronic hexecontahedron
• Small dodecahemicosacron
• Small dodecahemidodecacron
• Small dodecicosacron
• Small hexacronic icositetrahedron
• Small hexagonal hexecontahedron
• Small hexagrammic hexecontahedron
• Small icosacronic hexecontahedron
• Small icosihemidodecacron
• Small rhombidodecacron
• Small rhombihexacron
• Small stellapentakis dodecahedron
• Small triambic icosahedron
• Tetrahemihexacron

Honeycombs

Convex uniform honeycomb

• Cubic honeycomb
• Truncated cubic honeycomb
• Bitruncated cubic honeycomb
• Cantellated cubic honeycomb
• Cantitruncated cubic honeycomb
• Rectified cubic honeycomb
• Runcitruncated cubic honeycomb
• Omnitruncated cubic honeycomb
• Tetrahedral-octahedral honeycomb
• Truncated alternated cubic honeycomb
• Cantitruncated alternated cubic honeycomb
• Runcinated alternated cubic honeycomb
• Quarter cubic honeycomb
• Gyrated tetrahedral-octahedral honeycomb
• Gyrated triangular prismatic honeycomb
• Gyroelongated alternated cubic honeycomb
• Gyroelongated triangular prismatic honeycomb
• Elongated triangular prismatic honeycomb
• Elongated alternated cubic honeycomb
• Hexagonal prismatic honeycomb
• Triangular prismatic honeycomb
• Triangular-hexagonal prismatic honeycomb
• Truncated hexagonal prismatic honeycomb
• Truncated square prismatic honeycomb
• Rhombitriangular-hexagonal prismatic honeycomb
• Omnitruncated triangular-hexagonal prismatic honeycomb
• Snub triangular-hexagonal prismatic honeycomb
• Snub square prismatic honeycomb

Dual uniform honeycomb

• Disphenoid tetrahedral honeycomb
• Rhombic dodecahedral honeycomb

Others

• Trapezo-rhombic dodecahedral honeycomb
• Weaire–Phelan structure

Convex uniform honeycombs in hyperbolic space

• Order-4 dodecahedral honeycomb
• Order-5 cubic honeycomb
• Order-5 dodecahedral honeycomb
• Icosahedral honeycomb

Other

• Apeirogonal prism
• Apeirohedron
• Bicupola
• Cupola
• Bifrustum
• Boerdijk–Coxeter helix
• Császár polyhedron
• Flexible polyhedron
• Gyroelongated square bipyramid
• Heronian tetrahedron
• Hexagonal bifrustum
• Hexagonal truncated trapezohedron
• Hill tetrahedron
• Holyhedron
• Infinite skew polyhedron
• Jessen's icosahedron
• Near-miss Johnson solid
• Parallelepiped
• Pentagonal bifrustum
• Polytetrahedron
• Pyritohedron
• Rhombic enneacontahedron
• Rhombic icosahedron
• Rhombo-hexagonal dodecahedron
• Rhombohedron
• Scalenohedron
• Schönhardt polyhedron
• Square bifrustum
• Square truncated trapezohedron
• Szilassi polyhedron
• Tetradecahedron
• Tetradyakis hexahedron
• Tetrated dodecahedron
• Triangular bifrustum
• Triaugmented triangular prism
• Truncated rhombic dodecahedron
• Truncated trapezohedron
• Truncated triakis tetrahedron
• Tridyakis icosahedron
• Trigonal trapezohedron
• Regular skew polyhedron
• Waterman polyhedron
• Wedge

Regular and uniform compound polyhedra

Polyhedral compound and Uniform polyhedron compound

• Compound of cube and octahedron
• Compound of dodecahedron and icosahedron
• Compound of eight octahedra with rotational freedom
• Compound of eight triangular prisms
• Compound of five cubes
• Compound of five cuboctahedra
• Compound of five cubohemioctahedra
• Compound of five great cubicuboctahedra
• Compound of five great dodecahedra
• Compound of five great icosahedra
• Compound of five great rhombihexahedra
• Compound of five icosahedra
• Compound of five octahedra
• Compound of five octahemioctahedra
• Compound of five small cubicuboctahedra
• Compound of five small rhombicuboctahedra
• Compound of five small rhombihexahedra
• Compound of five small stellated dodecahedra
• Compound of five stellated truncated cubes
• Compound of five tetrahedra
• Compound of five tetrahemihexahedra
• Compound of five truncated cubes
• Compound of five truncated tetrahedra
• Compound of five uniform great rhombicuboctahedra
• Compound of four hexagonal prisms
• Compound of four octahedra
• Compound of four octahedra with rotational freedom
• Compound of four tetrahedra
• Compound of four triangular prisms
• Compound of great icosahedron and great stellated dodecahedron
• Compound of six cubes with rotational freedom
• Compound of six decagonal prisms
• Compound of six decagrammic prisms
• Compound of six pentagonal prisms
• Compound of six pentagrammic crossed antiprisms
• Compound of six pentagrammic prisms
• Compound of six tetrahedra
• Compound of six tetrahedra with rotational freedom
• Compound of small stellated dodecahedron and great dodecahedron
• Compound of ten hexagonal prisms
• Compound of ten octahedra
• Compound of ten tetrahedra
• Compound of ten triangular prisms
• Compound of ten truncated tetrahedra
• Compound of three cubes
• Compound of three tetrahedra
• Compound of twelve pentagonal antiprisms with rotational freedom
• Compound of twelve pentagonal prisms
• Compound of twelve pentagrammic prisms
• Compound of twelve tetrahedra with rotational freedom
• Compound of twenty octahedra
• Compound of twenty octahedra with rotational freedom
• Compound of twenty tetrahemihexahedra
• Compound of twenty triangular prisms
• Compound of two great dodecahedra
• Compound of two great icosahedra
• Compound of two great inverted snub icosidodecahedra
• Compound of two great retrosnub icosidodecahedra
• Compound of two great snub icosidodecahedra
• Compound of two icosahedra
• Compound of two inverted snub dodecadodecahedra
• Compound of two small stellated dodecahedra
• Compound of two snub cubes
• Compound of two snub dodecadodecahedra
• Compound of two snub dodecahedra
• Compound of two snub icosidodecadodecahedra
• Compound of two truncated tetrahedra
• Prismatic compound of antiprisms
• Prismatic compound of antiprisms with rotational freedom
• Prismatic compound of prisms
• Prismatic compound of prisms with rotational freedom

Four dimensions

Four-dimensional space

4-polytope – general term for a four dimensional polytope

Regular 4-polytope

• 5-cell, Tesseract, 16-cell, 24-cell, 120-cell, 600-cell

Abstract regular polytope

• 11-cell, 57-cell

Regular star 4-polytope

• Icosahedral 120-cell, Small stellated 120-cell, Great 120-cell, Grand 120-cell, Great stellated 120-cell, Grand stellated 120-cell, Great grand 120-cell, Great icosahedral 120-cell, Grand 600-cell, Great grand stellated 120-cell

Uniform 4-polytope

• Rectified 5-cell, Truncated 5-cell, Cantellated 5-cell, Runcinated 5-cell
• Rectified tesseract, Truncated tesseract, Cantellated tesseract, Runcinated tesseract
• Rectified 16-cell, Truncated 16-cell
• Rectified 24-cell, Truncated 24-cell, Cantellated 24-cell, Runcinated 24-cell, Snub 24-cell
• Rectified 120-cell, Truncated 120-cell, Cantellated 120-cell, Runcinated 120-cell
• Rectified 600-cell, Truncated 600-cell, Cantellated 600-cell

Prismatic uniform 4-polytope

• Grand antiprism
• Duoprism
• Tetrahedral prism, Truncated tetrahedral prism
• Truncated cubic prism, Truncated octahedral prism, Cuboctahedral prism, Rhombicuboctahedral prism, Truncated cuboctahedral prism, Snub cubic prism
• Truncated dodecahedral prism, Truncated icosahedral prism, Icosidodecahedral prism, Rhombicosidodecahedral prism, Truncated icosidodecahedral prism, Snub dodecahedral prism

Uniform antiprismatic prism

• Triangular antiprismatic prism, Square antiprismatic prism, Pentagonal antiprismatic prism, Hexagonal antiprismatic prism, Heptagonal antiprismatic prism, Octagonal antiprismatic prism, Enneagonal antiprismatic prism, Decagonal antiprismatic prism
• Pentagrammic antiprismatic prism, Hexagrammic antiprismatic prism, Heptagrammic antiprismatic prism, Octagrammic antiprismatic prism, Enneagrammic antiprismatic prism, Decagrammic antiprismatic prism
• Pentagrammic crossed antiprismatic prism, Hexagrammic crossed antiprismatic prism, Heptagrammic crossed antiprismatic prism, Octagrammic crossed antiprismatic prism, Enneagrammic crossed antiprismatic prism, Decagrammic crossed antiprismatic prism

Honeycombs

• Tesseractic honeycomb
• 24-cell honeycomb
• Snub 24-cell honeycomb
• Rectified 24-cell honeycomb
• Truncated 24-cell honeycomb
• 16-cell honeycomb
• 5-cell honeycomb
• Omnitruncated 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-simplex honeycomb

Five dimensions

Five-dimensional space, 5-polytope and uniform 5-polytope

• 5-simplex, Rectified 5-simplex, Truncated 5-simplex, Cantellated 5-simplex, Runcinated 5-simplex, Stericated 5-simplex
• 5-demicube, Truncated 5-demicube, Cantellated 5-demicube, Runcinated 5-demicube
• 5-cube, Rectified 5-cube, Truncated 5-cube, Cantellated 5-cube, Runcinated 5-cube, Stericated 5-cube
• 5-orthoplex, Rectified 5-orthoplex, Truncated 5-orthoplex, Cantellated 5-orthoplex, Runcinated 5-orthoplex

Prismatic uniform 5-polytope

• 5-cell prism, Rectified 5-cell prism, Truncated 5-cell prism, Cantellated 5-cell prism, Runcinated 5-cell prism, Bitruncated 5-cell prism, Cantitruncated 5-cell prism, Runcitruncated 5-cell prism, Omnitruncated 5-cell prism
• Tesseractic prism, Rectified tesseractic prism, Truncated tesseractic prism, Cantellated tesseractic prism, Runcinated tesseractic prism, Bitruncated tesseractic prism, Cantitruncated tesseractic prism, Runcitruncated tesseractic prism, Omnitruncated tesseractic prism
• 16-cell prism, Truncated 16-cell prism, Runcitruncated 16-cell prism
• 24-cell prism, rectified 24-cell prism, truncated 24-cell prism, cantellated 24-cell prism, runcinated 24-cell prism, bitruncated 24-cell prism, cantitruncated 24-cell prism, runcitruncated 24-cell prism, omnitruncated 24-cell prism, snub 24-cell prism
• 120-cell prism, Rectified 120-cell prism, Truncated 120-cell prism, Cantellated 120-cell prism, Runcinated 120-cell prism, Bitruncated 120-cell prism, Cantitruncated 120-cell prism, Runcitruncated 120-cell prism, Omnitruncated 120-cell prism
• 600-cell prism, Rectified 600-cell prism, Truncated 600-cell prism, Cantellated 600-cell prism, Cantitruncated 600-cell prism, Runcitruncated 600-cell prism
• Grand antiprism prism

Honeycombs

• 5-cubic honeycomb
• 5-simplex honeycomb
• Truncated 5-simplex honeycomb
• 5-demicubic honeycomb

Six dimensions

Six-dimensional space, 6-polytope and uniform 6-polytope

• 6-simplex, Rectified 6-simplex, Truncated 6-simplex, Cantellated 6-simplex, Runcinated 6-simplex, Stericated 6-simplex, Pentellated 6-simplex
• 6-demicube, Truncated 6-demicube, Cantellated 6-demicube, Runcinated 6-demicube, Stericated 6-demicube
• 6-cube, Rectified 6-cube, Truncated 6-cube, Cantellated 6-cube, Runcinated 6-cube, Stericated 6-cube, Pentellated 6-cube
• 6-orthoplex, Rectified 6-orthoplex, Truncated 6-orthoplex, Cantellated 6-orthoplex, Runcinated 6-orthoplex, Stericated 6-orthoplex
• 1₂₂ polytope, 2₂₁ polytope

Honeycombs

• 6-cubic honeycomb
• 6-simplex honeycomb
• 6-demicubic honeycomb
• 2₂₂ honeycomb

Seven dimensions

Seven-dimensional space, uniform 7-polytope

• 7-simplex, Rectified 7-simplex, Truncated 7-simplex, Cantellated 7-simplex, Runcinated 7-simplex, Stericated 7-simplex, Pentellated 7-simplex, Hexicated 7-simplex
• 7-demicube, Truncated 7-demicube, Cantellated 7-demicube, Runcinated 7-demicube, Stericated 7-demicube, Pentellated 7-demicube
• 7-cube, Rectified 7-cube, Truncated 7-cube, Cantellated 7-cube, Runcinated 7-cube, Stericated 7-cube, Pentellated 7-cube, Hexicated 7-cube
• 7-orthoplex, Rectified 7-orthoplex, Truncated 7-orthoplex, Cantellated 7-orthoplex, Runcinated 7-orthoplex, Stericated 7-orthoplex, Pentellated 7-orthoplex, Hexicated 7-orthoplex
• 1₃₂ polytope, 2₃₁ polytope, 3₂₁ polytope

Honeycombs

• 7-cubic honeycomb
• 7-demicubic honeycomb
• 3₃₁ honeycomb, 1₃₃ honeycomb

Eight dimension

Eight-dimensional space, uniform 8-polytope

• 8-simplex, Rectified 8-simplex, Truncated 8-simplex, Cantellated 8-simplex, Runcinated 8-simplex, Stericated 8-simplex, Pentellated 8-simplex, Hexicated 8-simplex, Heptellated 8-simplex
• 8-demicube, Truncated 8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube
• 8-cube, Rectified 8-cube, Truncated 8-cube, Cantellated 8-cube, Runcinated 8-cube, Stericated 8-cube, Pentellated 8-cube, Hexicated 8-cube, Heptellated 8-cube
• 8-orthoplex, Rectified 8-orthoplex, Truncated 8-orthoplex, Cantellated 8-orthoplex, Runcinated 8-orthoplex, Stericated 8-orthoplex, Pentellated 8-orthoplex, Hexicated 8-orthoplex,
• 1₄₂ polytope, 2₄₁ polytope, 4₂₁ polytope, Truncated 4₂₁ polytope, Truncated 2₄₁ polytope, Truncated 1₄₂ polytope, Cantellated 4₂₁ polytope, Cantellated 2₄₁ polytope, Runcinated 4₂₁ polytope

Honeycombs

• 8-cubic honeycomb
• 8-demicubic honeycomb
• 5₂₁ honeycomb, 2₅₁ honeycomb, 1₅₂ honeycomb

Nine dimensions

9-polytope

• 9-simplex
• 9-demicube
• 9-cube
• 9-orthoplex

Hyperbolic honeycombs

• E₉ honeycomb

Ten dimensions

10-polytope

• 10-simplex
• 10-demicube
• 10-cube
• 10-orthoplex

Dimensional families

Regular polytope and List of regular polytopes

• Simplex
• Hypercube
• Cross-polytope

Uniform polytope

• Demihypercube
• Uniform 1_k2 polytope
• Uniform 2_k1 polytope
• Uniform k₂₁ polytope

Honeycombs

• Hypercubic honeycomb
• Alternated hypercubic honeycomb

Geometric operators

• Rectification (geometry)
• Truncation (geometry)
• Bitruncation
• Cantellation
• Runcination
• Sterication
• Omnitruncation
• Expansion (geometry)
• Snub (geometry)
• Alternation (geometry)
• Dual polyhedron
• Gyration (geometry)
• Elongation (geometry)
• Augmentation (geometry)
• Diminishment (geometry)
• Greatening (geometry)
• Aggrandizement (geometry)
• Stellation
• Kleetope
• Conway polyhedron notation

See also

• List of geometry topics

Affine Coxeter groups

The affine Coxeter 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 Coxeter group, and the Coxeter graph of the affine Coxeter group is obtained from the Coxeter graph of the quotient group by adding another 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 (the affine symmetric group). For n = 2, this can be pictured as a subgroup of the symmetry group of the standard tiling of the plane by equilateral triangles.

In general, given a root system, one can construct the associated Stiefel diagram, consisting of the hyperplanes orthogonal to the roots along with certain translates of these hyperplanes. The affine Coxeter group (or affine Weyl group) is then the group generated by the (affine) reflections about all the hyperplanes in the diagram. The Stiefel diagram divides the plane into infinitely many connected components called alcoves, and the affine Coxeter group acts freely and transitively on the alcoves, just as the ordinary Weyl group acts freely and transitively on the Weyl chambers. The figure at right illustrates the Stiefel diagram for the ${\displaystyle G_{2}}$ root system.

Suppose ${\displaystyle R}$ is an irreducible root system of rank ${\displaystyle r>1}$ and let ${\displaystyle \alpha _{1},\ldots ,\alpha _{r}}$ be a collection of simple roots. Let, also, ${\displaystyle \alpha _{r+1}}$ denote the highest root. Then the affine Coxeter group is generated by the ordinary (linear) reflections about the hyperplanes perpendicular to ${\displaystyle \alpha _{1},\ldots ,\alpha _{r}}$, together with an affine reflection about a translate of the hyperplane perpendicular to ${\displaystyle \alpha _{r+1}}$. The Coxeter graph for the affine Weyl group is the Coxeter–Dynkin diagram for ${\displaystyle R}$, together with one additional node associated to ${\displaystyle \alpha _{r+1}}$. In this case, one alcove of the Stiefel diagram may be obtained by taking the fundamental Weyl chamber and cutting it by a translate of the hyperplane perpendicular to ${\displaystyle \alpha _{r+1}}$.

A list of the affine Coxeter groups follows:

Group symbol	Witt symbol	Bracket notation	Coxeter graph	Related uniform tessellation(s)
${\displaystyle {\tilde {A}}_{n}}$	${\displaystyle P_{n+1}}$		... or ...	Simplectic honeycomb
${\displaystyle {\tilde {B}}_{n}}$	${\displaystyle S_{n+1}}$		...	Demihypercubic honeycomb
${\displaystyle {\tilde {C}}_{n}}$	${\displaystyle R_{n+1}}$		...	Hypercubic honeycomb
${\displaystyle {\tilde {D}}_{n}}$	${\displaystyle Q_{n+1}}$		...	Demihypercubic honeycomb
${\displaystyle {\tilde {E}}_{6}}$	${\displaystyle T_{7}}$		or	222
${\displaystyle {\tilde {E}}_{7}}$	${\displaystyle T_{8}}$		or	331, 133
${\displaystyle {\tilde {E}}_{8}}$	${\displaystyle T_{9}}$			521, 251, 152
${\displaystyle {\tilde {F}}_{4}}$	${\displaystyle U_{5}}$			16-cell honeycomb 24-cell honeycomb
${\displaystyle {\tilde {G}}_{2}}$	${\displaystyle V_{3}}$			Hexagonal tiling and Triangular tiling
${\displaystyle {\tilde {A}}_{1}=I_{2}(\infty )}$	${\displaystyle W_{2}}$			Apeirogon

The group symbol 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 infinitely many hyperbolic Coxeter groups describing reflection groups in hyperbolic space, notably including the hyperbolic triangle groups.

Irreducible Coxeter groups

A Coxeter group is said to be irreducible if its Coxeter–Dynkin diagram is connected. Every Coxeter group is the direct product of the irreducible groups that correspond to the components of its Coxeter–Dynkin diagram.

Partial orders

A 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; this is precisely the length in the word metric in the Cayley graph. An expression for v using ℓ(v) generators is a reduced word. For example, the permutation (13) in S₃ has two reduced words, (12)(23)(12) and (23)(12)(23). The function ${\displaystyle v\to (-1)^{\ell (v)}}$ defines a map ${\displaystyle G\to \{\pm 1\},}$ generalizing the sign map for the symmetric group.

Using reduced words one may define three partial orders on the Coxeter group, the (right) weak order, the absolute order and the Bruhat order (named for François Bruhat). An element v exceeds an element u in the Bruhat order if some (or equivalently, any) reduced word for v contains a reduced word for u as a substring, where some letters (in any position) are dropped. In the weak order, v ≥ u if some reduced word for v contains a reduced word for u as an initial segment. Indeed, the word length makes this into a graded poset. The Hasse diagrams corresponding to these orders are objects of study, and are related to the Cayley graph determined by the generators. The absolute order is defined analogously to the weak order, but with generating set/alphabet consisting of all conjugates of the Coxeter generators.

For example, the permutation (1 2 3) in S₃ has only one reduced word, (12)(23), so covers (12) and (23) in the Bruhat order but only covers (12) in the weak order.

Homology

Since a Coxeter group ${\displaystyle W}$ is generated by finitely many elements of order 2, its abelianization is an elementary abelian 2-group, i.e., it is isomorphic to the direct sum of several copies of the cyclic group ${\displaystyle Z_{2}}$. This may be restated in terms of the first homology group of ${\displaystyle W}$.

The Schur multiplier ${\displaystyle M(W)}$, equal to the second homology group of ${\displaystyle W}$, was computed in (Ihara & Yokonuma 1965) for finite reflection groups and in (Yokonuma 1965) for affine reflection groups, with a more unified account given in (Howlett 1988). In all cases, the Schur multiplier is also an elementary abelian 2-group. For each infinite family ${\displaystyle \{W_{n}\}}$ of finite or affine Weyl groups, the rank of ${\displaystyle M(W_{n})}$ stabilizes as ${\displaystyle n}$ goes to infinity.

See also

• Artin–Tits group
• Chevalley–Shephard–Todd theorem
• Complex reflection group
• Coxeter element
• Iwahori–Hecke algebra, a quantum deformation of the group algebra
• Kazhdan–Lusztig polynomial
• Longest element of a Coxeter group
• Parabolic subgroup of a reflection group
• Supersolvable arrangement

Notes

1. In some contexts, the naming scheme may be extended to allow the following alternative or redundant names: ${\displaystyle B_{1}\cong A_{1}}$, ${\displaystyle D_{2}\cong I_{2}(2)\cong A_{1}\times A_{1}}$, ${\displaystyle I_{2}(3)\cong A_{2}}$, ${\displaystyle I_{2}(4)\cong B_{2}}$, ${\displaystyle H_{2}\cong I_{2}(5)}$, and ${\displaystyle D_{3}\cong A_{3}}$.
2. an index 2 subgroup of ${\displaystyle \operatorname {GO} _{4}^{+}(5)}$

References

1. ^ 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.
2. ^ Coxeter, H. S. M. (January 1935). "The complete enumeration of finite groups of the form ${\displaystyle r_{i}^{2}=(r_{i}r_{j})^{k_{ij}}=1}$". Journal of the London Mathematical Society: 21–25. doi:10.1112/jlms/s1-10.37.21.
3. Bourbaki, Nicolas (2002). "4-6". Lie Groups and Lie Algebras. Elements of Mathematics. Springer. ISBN 978-3-540-42650-9. Zbl 0983.17001.
4. 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.
5. 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.
6. Brink, Brigitte; Howlett, Robert B. (1993). "A finiteness property and an automatic structure for Coxeter groups". Mathematische Annalen. 296 (1): 179–190. doi:10.1007/BF01445101. S2CID 122177473. Zbl 0793.20036.
7. Coxeter, H. S. M. (January 1973). "12.6. The number of reflections". Regular Polytopes. Courier Corporation. ISBN 0-486-61480-8.
8. Wilson, Robert A. (2009), "Chapter 2", The finite simple groups, Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5
9. Hall 2015 Section 13.6
10. Hall 2015 Chapter 13, Exercises 12 and 13

Bibliography

• Hall, Brian C. (2015). Lie groups, Lie algebras, and representations: An elementary introduction. Graduate Texts in Mathematics. Vol. 222 (2nd ed.). Springer. ISBN 978-3-319-13466-6.
• Ihara, S.; Yokonuma, Takeo (1965). "On the second cohomology groups (Schur-multipliers) of finite reflection groups" (PDF). J. Fac. Sci. Univ. Tokyo, Sect. 1. 11: 155–171. Zbl 0136.28802. Archived from the original (PDF) on 2013-10-23.
• Howlett, Robert B. (1988). "On the Schur Multipliers of Coxeter Groups". J. London Math. Soc. 2. 38 (2): 263–276. doi:10.1112/jlms/s2-38.2.263. Zbl 0627.20019.
• Yokonuma, Takeo (1965). "On the second cohomology groups (Schur-multipliers) of infinite discrete reflection groups". J. Fac. Sci. Univ. Tokyo, Sect. 1. 11: 173–186. hdl:2261/6049. Zbl 0136.28803.

Further reading

• Björner, Anders; Brenti, Francesco (2005). Combinatorics of Coxeter Groups. Graduate Texts in Mathematics. Vol. 231. Springer. ISBN 978-3-540-27596-1. Zbl 1110.05001.
• Grove, Larry C.; Benson, Clark T. (1985). Finite Reflection Groups. Graduate texts in mathematics. Vol. 99. Springer. ISBN 978-0-387-96082-1.
• Kane, Richard (2001). Reflection Groups and Invariant Theory. CMS Books in Mathematics. Springer. ISBN 978-0-387-98979-2. Zbl 0986.20038.
• Hiller, Howard (1982). Geometry of Coxeter groups. Research Notes in Mathematics. Vol. 54. Pitman. ISBN 978-0-273-08517-1. Zbl 0483.57002.
• Vinberg, Ernest B. (1984). "Absence of crystallographic groups of reflections in Lobachevski spaces of large dimension". Trudy Moskov. Mat. Obshch. 47.

External links

• "Coxeter group", Encyclopedia of Mathematics, EMS Press, 2001
• Weisstein, Eric W., "Coxeter group", MathWorld
• Jenn software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators

• Polyhedra
• Polygons
• Polytopes
• Lists of shapes
• Coxeter groups
• Reflection groups