Inmathematics, aCoxeter group, named afterH. S. M. Coxeter, is anabstract group that admits aformal description in terms ofreflections (orkaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclideanreflection groups; for example, thesymmetry group of eachregular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms ofsymmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups,^[1] and finite Coxeter groups were classified in 1935.^[2]

Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups ofregular polytopes, and theWeyl groups ofsimple Lie algebras. Examples of infinite Coxeter groups include thetriangle groups corresponding toregular tessellations of theEuclidean plane and thehyperbolic plane, and the Weyl groups of infinite-dimensionalKac–Moody algebras.^[3][4][5]

Formally, aCoxeter group can be defined as a group with thepresentation

${\displaystyle ⟨r_1,r_2,\dots ,r_n\mid (r_i{r}_j{)}^{m_{ij}}=1⟩}$

where${\displaystyle m_{ii}=1}$ and${\displaystyle m_{ij}=m_{ji}\ge 2}$ is either an integer or${\displaystyle \mathrm{\infty }}$ for${\displaystyle i\ne j}$.Here, the condition${\displaystyle m_{ij}=\mathrm{\infty }}$ means that no relation of the form${\displaystyle (r_i{r}_j{)}^m=1}$ for any integer${\displaystyle m\ge 2}$ 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 aCoxeter system. Note that in general${\displaystyle S}$ isnot 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 areinvolutions.
• 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, thecommutator 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${\displaystyle i\ne j}$ 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${\displaystyle (yx{)}^k}$ areconjugates: indeed${\displaystyle y(xy{)}^k{y}^{-1}=(yx{)}^{k}y{y}^{-1}=(yx{)}^k}$.

TheCoxeter 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${\displaystyle \{2,3,\dots \}\cup \{\mathrm{\infty }\}}$ is a Coxeter matrix.

The Coxeter matrix can be conveniently encoded by aCoxeter 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${\displaystyle m_{ij}\ge 3}$.
• An edge is labelled with the value of${\displaystyle m_{ij}}$ whenever the value is${\displaystyle 4}$ or greater.

In particular, two generatorscommute if and only if they are not joined by an edge. Furthermore, if a Coxeter graph has two or moreconnected components, the associated group is thedirect product of the groups associated to the individual components.Thus thedisjoint union of Coxeter graphs yields adirect 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 thedot product of the pairwise generators. The Schläfli matrix is useful because itseigenvalues determine whether the Coxeter group is offinite type (all positive),affine type (all non-negative, at least one zero), orindefinite 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	${\displaystyle {\tilde{A}}_1=I_2(\mathrm{\infty })}$	A3	B3	D4	${\displaystyle {\tilde{A}}_3}$
Coxeter diagram
Coxeter matrix
Schläfli matrix

The graph${\displaystyle A_n}$ in whichvertices${\displaystyle 1}$ through${\displaystyle n}$ are placed in a row with each vertex joined by an unlabellededge to its immediate neighbors is the Coxeter diagram of thesymmetric group${\displaystyle S_{n+1}}$; thegenerators correspond to thetranspositions${\displaystyle (12),(23),\dots ,(nn+1)}$. Any two non-consecutive transpositions commute, while multiplying two consecutive transpositions gives a 3-cycle :${\displaystyle (kk+1)\cdot (k+1k+2)=(kk+2k+1)}$. Therefore${\displaystyle S_{n+1}}$ is aquotient of the Coxeter group having Coxeter diagram${\displaystyle A_n}$. Further arguments show that this quotient map is an isomorphism.

Coxeter groups are an abstraction of reflection groups. Coxeter groups areabstract groups, in the sense of being given via a presentation. On the other hand, reflection groups areconcrete, 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 alinear 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 twohyperplanes 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 orderk.

In this way, every reflection group may be presented as a Coxeter group.^[1] The converse is partially true: every finite Coxeter group admits a faithfulrepresentation as a finite reflection group of some Euclidean space.^[2]However, not every infinite Coxeter group admits a representation as a reflection group.

Finite Coxeter groups have been classified.^[2]

Finite Coxeter groups are classified in terms of theirCoxeter diagrams.^[2]

The finite Coxeter groups with connected Coxeter diagrams consist of three one-parameter families of increasing dimension (${\displaystyle A_n}$ for${\displaystyle n\ge 1}$,${\displaystyle B_n}$ for${\displaystyle n\ge 2}$, and${\displaystyle D_n}$ for${\displaystyle n\ge 4}$), a one-parameter family of dimension two (${\displaystyle I_2(p)}$ for${\displaystyle p\ge 5}$), and sixexceptional groups (${\displaystyle E_6,E_7,E_8,F_4,H_3,}$ and${\displaystyle H_4}$). Every finite Coxeter group is thedirect product of finitely many of these irreducible groups.^[a]

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 notexceptionally 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 anautomatic group.^[6] 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 thecrystallographic 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${\displaystyle I_2(p)}$ ap-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 diagramsB_n andC_n give rise to the same Weyl group (hence Coxeter group), because they differ asdirected graphs, but agree asundirected graphs – direction matters for root systems but not for the Weyl group; this corresponds to thehypercube andcross-polytope being different regular polytopes but having the same symmetry group.

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.

Rankn	Group symbol	Alternate symbol	Bracket notation	Coxeter graph	Reflections m =⁠1/2⁠nh[7]	Coxeter number h	Order	Group structure[8]	Relatedpolytopes
1	A1	A1	[ ]		1	2	2	${\displaystyle S_2}$	{ }
2	A2	A2	[3]		3	3	6	${\displaystyle S_3\cong D_6\cong {\mathrm{GO}}_2^{-}(2)\cong {\mathrm{GO}}_2^{+}(4)}$	{3}
3	A3	A3	[3,3]		6	4	24	${\displaystyle S_4}$	{3,3}
4	A4	A4	[3,3,3]		10	5	120	${\displaystyle S_5}$	{3,3,3}
5	A5	A5	[3,3,3,3]		15	6	720	${\displaystyle S_6}$	{3,3,3,3}
n	An	An	[3n−1]	...	n(n + 1)/2	n + 1	(n + 1)!	${\displaystyle S_{n+1}}$	n-simplex
2	B2	C2	[4]		4	4	8	${\displaystyle C_2\wr S_2\cong D_8\cong {\mathrm{GO}}_2^{-}(3)\cong {\mathrm{GO}}_2^{+}(5)}$	{4}
3	B3	C3	[4,3]		9	6	48	${\displaystyle C_2\wr S_3\cong S_4\times 2}$	{4,3} /{3,4}
4	B4	C4	[4,3,3]		16	8	384	${\displaystyle C_2\wr S_4}$	{4,3,3} /{3,3,4}
5	B5	C5	[4,3,3,3]		25	10	3840	${\displaystyle C_2\wr S_5}$	{4,3,3,3} /{3,3,3,4}
n	Bn	Cn	[4,3n−2]	...	n2	2n	2nn!	${\displaystyle C_2\wr S_n}$	n-cube /n-orthoplex
4	D4	B4	[31,1,1]		12	6	192	${\displaystyle C_2^3{S}_4\cong 2^{1+4}:S_3}$	h{4,3,3} /{3,31,1}
5	D5	B5	[32,1,1]		20	8	1920	${\displaystyle C_2^4{S}_5}$	h{4,3,3,3} /{3,3,31,1}
n	Dn	Bn	[3n−3,1,1]	...	n(n − 1)	2(n − 1)	2n−1n!	${\displaystyle C_2^{n-1}{S}_n}$	n-demicube /n-orthoplex
6	E6	E6	[32,2,1]		36	12	51840	${\displaystyle {\mathrm{GO}}_6^{-}(2)\cong {\mathrm{SO}}_5(3)\cong {\mathrm{PSp}}_4(3):2\cong {\mathrm{PSU}}_4(2):2}$	221,122
7	E7	E7	[33,2,1]		63	18	2903040	${\displaystyle {\mathrm{GO}}_7(2)\times 2\cong {\mathrm{Sp}}_6(2)\times 2}$	321,231,132
8	E8	E8	[34,2,1]		120	30	696729600	${\displaystyle 2\cdot {\mathrm{GO}}_8^{+}(2)}$	421,241,142
4	F4	F4	[3,4,3]		24	12	1152	${\displaystyle {\mathrm{GO}}_4^{+}(3)\cong 2^{1+4}:(S_3\times S_3)}$	{3,4,3}
2	G2	– (D6 2)	[6]		6	6	12	${\displaystyle D_{12}\cong {\mathrm{GO}}_2^{-}(5)\cong {\mathrm{GO}}_2^{+}(7)}$	{6}
2	I2(5)	G2	[5]		5	5	10	${\displaystyle D_{10}\cong {\mathrm{GO}}_2^{-}(4)}$	{5}
3	H3	G3	[3,5]		15	10	120	${\displaystyle 2\times A_5}$	{3,5} /{5,3}
4	H4	G4	[3,3,5]		60	30	14400	${\displaystyle 2\cdot (A_5\times A_5):2}$[b]	{5,3,3} /{3,3,5}
2	I2(n)	Dn 2	[n]		n	n	2n	${\displaystyle D_{2n}}$ ${\displaystyle \cong {\mathrm{GO}}_2^{-}(n-1)}$ whenn =pk + 1,p prime ${\displaystyle \cong {\mathrm{GO}}_2^{+}(n+1)}$ whenn =pk − 1,p prime	{p}

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

There are three series of regular polytopes in all dimensions. The symmetry group of a regularn-simplex is the symmetric groupS_n+1, also known as the Coxeter group of typeA_n. The symmetry group of then-cube and its dual, then-cross-polytope, isB_n, and is known as thehyperoctahedral group.

The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter groups. In two dimensions, thedihedral groups, which are the symmetry groups ofregular polygons, form the seriesI₂(p), forp ≥ 3. In three dimensions, the symmetry group of the regulardodecahedron and its dual, the regularicosahedron, isH₃, known as thefull icosahedral group. In four dimensions, there are three exceptional regular polytopes, the24-cell, the120-cell, and the600-cell. The first has symmetry groupF₄, while the other two are dual and have symmetry groupH₄.

The Coxeter groups of typeD_n,E₆,E₇, andE₈ are the symmetry groups of certainsemiregular polytopes.

Theaffine Coxeter groups form a second important series of Coxeter groups. These are not finite themselves, but each contains anormalabeliansubgroup 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, forn ≥ 2, the graph consisting ofn+1 vertices in a circle is obtained fromA_n in this way, and the corresponding Coxeter group is the affine Weyl group ofA_n (theaffine symmetric group). Forn = 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 associatedStiefel 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.^[9] The Stiefel diagram divides the plane into infinitely many connected components calledalcoves, 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${\displaystyle {\alpha }_1,\dots ,{\alpha }_r}$ 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${\displaystyle {\alpha }_1,\dots ,{\alpha }_r}$, 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${\displaystyle {\alpha }_{r+1}}$.^[10]

A list of the affine Coxeter groups follows:

Group symbol	Witt symbol	Bracket notation	Coxeter graph	Related uniform tessellation(s)
${\displaystyle {\tilde{A}}_n}$	${\displaystyle P_{n+1}}$	[3[n+1]]	... or ...	Simplectic honeycomb
${\displaystyle {\tilde{B}}_n}$	${\displaystyle S_{n+1}}$	[4,3n − 3,31,1]	...	Demihypercubic honeycomb
${\displaystyle {\tilde{C}}_n}$	${\displaystyle R_{n+1}}$	[4,3n−2,4]	...	Hypercubic honeycomb
${\displaystyle {\tilde{D}}_n}$	${\displaystyle Q_{n+1}}$	[ 31,1,3n−4,31,1]	...	Demihypercubic honeycomb
${\displaystyle {\tilde{E}}_6}$	${\displaystyle T_7}$	[32,2,2]	or	222
${\displaystyle {\tilde{E}}_7}$	${\displaystyle T_8}$	[33,3,1]	or	331,133
${\displaystyle {\tilde{E}}_8}$	${\displaystyle T_9}$	[35,2,1]		521,251,152
${\displaystyle {\tilde{F}}_4}$	${\displaystyle U_5}$	[3,4,3,3]		16-cell honeycomb 24-cell honeycomb
${\displaystyle {\tilde{G}}_2}$	${\displaystyle V_3}$	[6,3]		Hexagonal tiling and Triangular tiling
${\displaystyle {\tilde{A}}_1=I_2(\mathrm{\infty })}$	${\displaystyle W_2}$	[∞]		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.

There are infinitely manyhyperbolic Coxeter groups describing reflection groups inhyperbolic space, notably including the hyperbolic triangle groups.

A Coxeter group is said to beirreducible if its Coxeter–Dynkin diagram is connected. Every Coxeter group is thedirect product of the irreducible groups that correspond to thecomponents of its Coxeter–Dynkin diagram.

A choice of reflection generators gives rise to alength functionℓ on a Coxeter group, namely the minimum number of uses of generators required to express a group element; this is precisely the length in theword metric in theCayley graph. An expression forv usingℓ(v) generators is areduced word. For example, the permutation (13) inS₃ 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 thesign map for the symmetric group.

Using reduced words one may define threepartial orders on the Coxeter group, the (right)weak order, theabsolute order and theBruhat order (named forFrançois Bruhat). An elementv exceeds an elementu in the Bruhat order if some (or equivalently, any) reduced word forv contains a reduced word foru as a substring, where some letters (in any position) are dropped. In the weak order,v ≥ u if some reduced word forv contains a reduced word foru as an initial segment. Indeed, the word length makes this into agraded poset. TheHasse diagrams corresponding to these orders are objects of study, and are related to theCayley 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) inS₃ has only one reduced word, (12)(23), so covers (12) and (23) in the Bruhat order but only covers (12) in the weak order.

Since a Coxeter group${\displaystyle W}$ is generated by finitely many elements of order 2, itsabelianization is anelementary abelian 2-group, i.e., it is isomorphic to the direct sum of several copies of thecyclic group${\displaystyle Z_2}$. This may be restated in terms of the firsthomology group of${\displaystyle W}$.

TheSchur 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.

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

• "Coxeter group",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Weisstein, Eric W.,"Coxeter group",MathWorld
• Jenn software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators

• Coxeter groups
• Reflection groups

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