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In mathematics, a Coxeter
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; the symmetry groups of regular polyhedra are an example. However, not all Coxeter
Coxeter
groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter
Coxeter
groups were introduced ( Coxeter
Coxeter
1934) as abstractions of reflection groups, and finite Coxeter
Coxeter
groups were classified in 1935 ( Coxeter
Coxeter
1935). Coxeter
Coxeter
groups find applications in many areas of mathematics. Examples of finite Coxeter
Coxeter
groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter
Coxeter
groups include the triangle groups corresponding to regular tessellations of the Euclidean plane
Euclidean plane
and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. Standard references include (Humphreys 1992) and (Davis 2007).

Contents

1 Definition

1.1 Coxeter
Coxeter
matrix and Schläfli matrix

2 An example 3 Connection with reflection groups 4 Finite Coxeter
Coxeter
groups

4.1 Classification 4.2 Weyl groups 4.3 Properties 4.4 Symmetry groups of regular polytopes

5 Affine Coxeter
Coxeter
groups 6 Hyperbolic Coxeter
Coxeter
groups 7 Partial orders 8 Homology 9 See also 10 References 11 Further reading 12 External links

Definition[edit] Formally, a Coxeter
Coxeter
group can be defined as a group with the presentation

r

1

,

r

2

, … ,

r

n

∣ (

r

i

r

j

)

m

i j

= 1

displaystyle leftlangle r_ 1 ,r_ 2 ,ldots ,r_ n mid (r_ i r_ j )^ m_ ij =1rightrangle

where

m

i i

= 1

displaystyle m_ ii =1

and

m

i j

≥ 2

displaystyle m_ ij geq 2

for

i ≠ j

displaystyle ineq j

. The condition

m

i j

= ∞

displaystyle m_ ij =infty

means no relation of the form

(

r

i

r

j

)

m

displaystyle (r_ i r_ j )^ m

should be imposed. The pair

( W , S )

displaystyle (W,S)

where

W

displaystyle W

is a Coxeter
Coxeter
group with generators

S =

r

1

, … ,

r

n

displaystyle S= r_ 1 ,dots ,r_ n

is called a Coxeter
Coxeter
system. Note that in general

S

displaystyle S

is not uniquely determined by

W

displaystyle W

. For example, the Coxeter
Coxeter
groups of type

B

3

displaystyle B_ 3

and

A

1

×

A

3

displaystyle A_ 1 times A_ 3

are isomorphic but the Coxeter
Coxeter
systems are not equivalent (see below for an explanation of this notation). A number of conclusions can be drawn immediately from the above definition.

The relation

m

i i

= 1

displaystyle m_ ii =1

means that

(

r

i

r

i

)

1

= (

r

i

)

2

= 1

displaystyle (r_ i r_ i )^ 1 =(r_ i )^ 2 =1

for all

i

displaystyle i

 ; as such the generators are involutions. If

m

i j

= 2

displaystyle m_ ij =2

, then the generators

r

i

displaystyle r_ i

and

r

j

displaystyle r_ j

commute. This follows by observing that

x x = y y = 1

displaystyle xx=yy=1

,

together with

x y x y = 1

displaystyle xyxy=1

implies that

x y = x ( x y x y ) y = ( x x ) y x ( y y ) = y x

displaystyle xy=x(xyxy)y=(xx)yx(yy)=yx

.

Alternatively, since the generators are involutions,

r

i

=

r

i

− 1

displaystyle r_ i =r_ i ^ -1

, so

(

r

i

r

j

)

2

=

r

i

r

j

r

i

r

j

=

r

i

r

j

r

i

− 1

r

j

− 1

displaystyle (r_ i r_ j )^ 2 =r_ i r_ j r_ i r_ j =r_ i r_ j r_ i ^ -1 r_ j ^ -1

, and thus is equal to the commutator.

In order to avoid redundancy among the relations, it is necessary to assume that

m

i j

=

m

j i

displaystyle m_ ij =m_ ji

. This follows by observing that

y y = 1

displaystyle yy=1

,

together with

( x y

)

m

= 1

displaystyle (xy)^ m =1

implies that

( y x

)

m

= ( y x

)

m

y y = y ( x y

)

m

y = y y = 1

displaystyle (yx)^ m =(yx)^ m yy=y(xy)^ m y=yy=1

.

Alternatively,

( x y

)

k

displaystyle (xy)^ k

and

( y x

)

k

displaystyle (yx)^ k

are conjugate elements, as

y ( x y

)

k

y

− 1

= ( y x

)

k

y

y

− 1

= ( y x

)

k

displaystyle y(xy)^ k y^ -1 =(yx)^ k yy^ -1 =(yx)^ k

.

Coxeter
Coxeter
matrix and Schläfli matrix[edit] The Coxeter
Coxeter
matrix is the

n × n

displaystyle ntimes n

, symmetric matrix with entries

m

i j

displaystyle m_ ij

. Indeed, every symmetric matrix with positive integer and ∞ entries and with 1's on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter
Coxeter
group. The Coxeter
Coxeter
matrix can be conveniently encoded by a Coxeter
Coxeter
diagram, as per the following rules.

The vertices of the graph are labelled by generator subscripts. Vertices

i

displaystyle i

and

j

displaystyle j

are adjacent if and only if

m

i j

≥ 3

displaystyle m_ ij geq 3

. An edge is labelled with the value of

m

i j

displaystyle m_ ij

whenever the value is

4

displaystyle 4

or greater.

In particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter
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
Coxeter
graphs yields a direct product of Coxeter
Coxeter
groups. The Coxeter
Coxeter
matrix,

M

i j

displaystyle M_ ij

, is related to the

n × n

displaystyle ntimes n

Schläfli matrix
Schläfli matrix

C

displaystyle C

with entries

C

i j

= − 2 cos ⁡ ( π

/

M

i j

)

displaystyle C_ ij =-2cos(pi /M_ ij )

, but the elements are modified, being proportional to the dot product of the pairwise generators. The Schläfli matrix
Schläfli matrix
is useful because its eigenvalues determine whether the Coxeter
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
Coxeter
groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups.

Examples

Coxeter
Coxeter
group A1×A1 A2

I ~

1

displaystyle tilde I _ 1

A3 B3 D4

A ~

3

displaystyle tilde A _ 3

Coxeter
Coxeter
diagram

Coxeter
Coxeter
matrix

[

1

2

2

1

]

displaystyle left[ begin smallmatrix 1&2\2&1\end smallmatrix right]

[

1

3

3

1

]

displaystyle left[ begin smallmatrix 1&3\3&1\end smallmatrix right]

[

1

1

]

displaystyle left[ begin smallmatrix 1&infty \infty &1\end smallmatrix right]

[

1

3

2

3

1

3

2

3

1

]

displaystyle left[ begin smallmatrix 1&3&2\3&1&3\2&3&1end smallmatrix right]

[

1

4

2

4

1

3

2

3

1

]

displaystyle left[ begin smallmatrix 1&4&2\4&1&3\2&3&1end smallmatrix right]

[

1

3

2

2

3

1

3

3

2

3

1

2

2

3

2

1

]

displaystyle left[ begin smallmatrix 1&3&2&2\3&1&3&3\2&3&1&2\2&3&2&1end smallmatrix right]

[

1

3

2

3

3

1

3

2

2

3

1

3

3

2

3

1

]

displaystyle left[ begin smallmatrix 1&3&2&3\3&1&3&2\2&3&1&3\3&2&3&1end smallmatrix right]

Schläfli matrix

[

2

0

0

2

]

displaystyle left[ begin smallmatrix 2&0\0&2end smallmatrix right]

[

 

2

− 1

− 1

 

2

]

displaystyle left[ begin smallmatrix ,2&-1\-1& ,2end smallmatrix right]

[

 

2

− 2

− 2

 

2

]

displaystyle left[ begin smallmatrix ,2&-2\-2& ,2end smallmatrix right]

[

 

2

− 1

 

0

− 1

 

2

− 1

 

0

− 1

 

2

]

displaystyle left[ begin smallmatrix ,2&-1& ,0\-1& ,2&-1\ ,0&-1& ,2end smallmatrix right]

[

 

    2

2

 

0

2

 

    2

− 1

 

    0

 

− 1

 

2

]

displaystyle left[ begin smallmatrix , 2&- sqrt 2 & ,0\- sqrt 2 & , 2&-1\ , 0& ,-1& ,2end smallmatrix right]

[

 

2

− 1

 

0

 

0

− 1

 

2

− 1

− 1

 

0

− 1

 

2

 

0

 

0

− 1

 

0

 

2

]

displaystyle left[ begin smallmatrix ,2&-1& ,0& ,0\-1& ,2&-1&-1\ ,0&-1& ,2& ,0\ ,0&-1& ,0& ,2end smallmatrix right]

[

 

2

− 1

 

0

− 1

− 1

 

2

− 1

 

0

 

0

− 1

 

2

− 1

− 1

 

0

− 1

 

2

]

displaystyle left[ begin smallmatrix ,2&-1& ,0&-1\-1& ,2&-1& ,0\ ,0&-1& ,2&-1\-1& ,0&-1& ,2end smallmatrix right]

An example[edit] 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 Sn+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+2 k+1). Of course this only shows that Sn+1 is a quotient group of the Coxeter
Coxeter
group described by the graph, but it is not too difficult to check that equality holds. Connection with reflection groups[edit] Further information: Reflection group Coxeter
Coxeter
groups are deeply connected with reflection groups. Simply put, Coxeter
Coxeter
groups are abstract groups (given via a presentation), while reflection groups are concrete groups (given as subgroups of linear groups or various generalizations). Coxeter
Coxeter
groups grew out of the study of reflection groups — they are an abstraction: a reflection group is a subgroup of a linear group generated by reflections (which have order 2), while a Coxeter
Coxeter
group is an abstract group generated by involutions (elements of order 2, abstracting from reflections), and whose relations have a certain form (

(

r

i

r

j

)

k

displaystyle (r_ i r_ j )^ k

, corresponding to hyperplanes meeting at an angle of

π

/

k

displaystyle pi /k

, with

r

i

r

j

displaystyle r_ i r_ j

being of order k abstracting from a rotation by

2 π

/

k

displaystyle 2pi /k

). The abstract group of a reflection group is a Coxeter
Coxeter
group, while conversely a reflection group can be seen as a linear representation of a Coxeter
Coxeter
group. For finite reflection groups, this yields an exact correspondence: every finite Coxeter
Coxeter
group admits a faithful representation as a finite reflection group of some Euclidean space. For infinite Coxeter
Coxeter
groups, however, a Coxeter
Coxeter
group may not admit a representation as a reflection group. Historically, ( Coxeter
Coxeter
1934) proved that every reflection group is a Coxeter
Coxeter
group (i.e., has a presentation where all relations are of the form

r

i

2

displaystyle r_ i ^ 2

or

(

r

i

r

j

)

k

displaystyle (r_ i r_ j )^ k

), and indeed this paper introduced the notion of a Coxeter
Coxeter
group, while ( Coxeter
Coxeter
1935) proved that every finite Coxeter
Coxeter
group had a representation as a reflection group, and classified finite Coxeter groups. Finite Coxeter
Coxeter
groups[edit]

Coxeter
Coxeter
graphs of the finite Coxeter
Coxeter
groups.

Classification[edit] The finite Coxeter
Coxeter
groups were classified in ( Coxeter
Coxeter
1935), in terms of Coxeter–Dynkin diagrams; they are all represented by reflection groups of finite-dimensional Euclidean spaces. The finite Coxeter
Coxeter
groups consist of three one-parameter families of increasing rank

A

n

,

B

n

,

D

n

,

displaystyle A_ n ,B_ n ,D_ n ,

one one-parameter family of dimension two,

I

2

( p ) ,

displaystyle I_ 2 (p),

and six exceptional groups:

E

6

,

E

7

,

E

8

,

F

4

,

H

3

,

displaystyle E_ 6 ,E_ 7 ,E_ 8 ,F_ 4 ,H_ 3 ,

and

H

4

.

displaystyle H_ 4 .

Weyl groups[edit] Main article: Weyl group Many, but not all of these, are Weyl groups, and every Weyl group
Weyl group
can be realized as a Coxeter
Coxeter
group. The Weyl groups are the families

A

n

,

B

n

,

displaystyle A_ n ,B_ n ,

and

D

n

,

displaystyle D_ n ,

and the exceptions

E

6

,

E

7

,

E

8

,

F

4

,

displaystyle E_ 6 ,E_ 7 ,E_ 8 ,F_ 4 ,

and

I

2

( 6 ) ,

displaystyle I_ 2 (6),

denoted in Weyl group
Weyl group
notation as

G

2

.

displaystyle G_ 2 .

The non-Weyl groups are the exceptions

H

3

displaystyle H_ 3

and

H

4

,

displaystyle H_ 4 ,

and the family

I

2

( p )

displaystyle I_ 2 (p)

except where this coincides with one of the Weyl groups (namely

I

2

( 3 ) ≅

A

2

,

I

2

( 4 ) ≅

B

2

,

displaystyle I_ 2 (3)cong A_ 2 ,I_ 2 (4)cong B_ 2 ,

and

I

2

( 6 ) ≅

G

2

displaystyle I_ 2 (6)cong G_ 2

). This can be proven by comparing the restrictions on (undirected) Dynkin diagrams with the restrictions on Coxeter
Coxeter
diagrams of finite groups: formally, the Coxeter
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
Coxeter
group is an automatic group.[1] 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

H

3

,

displaystyle H_ 3 ,

the dodecahedron (dually, icosahedron) does not fill space; for

H

4

,

displaystyle H_ 4 ,

the 120-cell
120-cell
(dually, 600-cell) does not fill space; for

I

2

( p )

displaystyle I_ 2 (p)

a p-gon does not tile the plane except for

p = 3 , 4 ,

displaystyle p=3,4,

or

6

displaystyle 6

(the triangular, square, and hexagonal tilings, respectively). Note further that the (directed) Dynkin diagrams Bn and Cn give rise to the same Weyl group
Weyl group
(hence Coxeter
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[edit] Some properties of the finite irreducible Coxeter
Coxeter
groups are given in the following table. The order of reducible groups can be computed by the product of their irreducible subgroup orders.

Rank n Group symbol Alternate symbol Bracket notation Coxeter graph Reflections m=nh/2[2] Coxeter
Coxeter
number h Order Related polytopes

1 A1 A1 [ ]

1 2 2

2 A2 A2 [3]

3 3 6 3

3 A3 A3 [3,3]

6 4 24 3,3

4 A4 A4 [3,3,3]

10 5 120 3,3,3

5 A5 A5 [3,3,3,3]

15 6 720 3,3,3,3

6 A6 A6 [3,3,3,3,3]

21 7 5040 3,3,3,3,3

7 A7 A7 [3,3,3,3,3,3]

28 8 40320 3,3,3,3,3,3

8 A8 A8 [3,3,3,3,3,3,3]

36 9 362880 3,3,3,3,3,3,3

n An An [3n-1] ... n(n+1)/2 n+1 (n + 1)! n-simplex

2 B2 C2 [4]

4 4 8 4

3 B3 C3 [4,3]

9 6 48 4,3 / 3,4

4 B4 C4 [4,3,3]

16 8 384 - 4,3,3 / 3,3,4

5 B5 C5 [4,3,3,3]

25 10 3840 4,3,3,3 / 3,3,3,4

6 B6 C6 [4,3,3,3,3]

36 12 46080 4,3,3,3,3 / 3,3,3,3,4

7 B7 C7 [4,3,3,3,3,3]

49 14 645120 4,3,3,3,3,3 / 3,3,3,3,3,4

8 B8 C8 [4,3,3,3,3,3,3]

64 16 10321920 4,3,3,3,3,3 / 3,3,3,3,3,4

n Bn Cn [4,3n-2] ... n2 2n 2n n! n-cube / n-orthoplex

4 D4 B4 [31,1,1]

12 6 192 h 4,3,3 / 3,31,1

5 D5 B5 [32,1,1]

20 8 1920 h 4,3,3,3 / 3,3,31,1

6 D6 B6 [33,1,1]

30 10 23040 h 4,3,3,3,3 / 3,3,3,31,1

7 D7 B7 [34,1,1]

42 12 322560 h 4,3,3,3,3,3 / 3,3,3,3,31,1

8 D8 B8 [35,1,1]

56 14 5160960 h 4,3,3,3,3,3,3 / 3,3,3,3,3,31,1

n Dn Bn [3n-3,1,1] ... n(n-1) 2(n-1) 2n−1 n! n-demicube / n-orthoplex

6 E6 E6 [32,2,1]

36 12 51840 (72x6!) 221, 122

7 E7 E7 [33,2,1]

63 18 2903040 (72x8!) 321, 231, 132

8 E8 E8 [34,2,1]

120 30 696729600 (192x10!) 421, 241, 142

4 F4 F4 [3,4,3]

24 12 1152 3,4,3

2 G2 - [6]

6 6 12 6

2 H2 G2 [5]

5 5 10 5

3 H3 G3 [3,5]

15 10 120 3,5 / 5,3

4 H4 G4 [3,3,5]

60 30 14400 5,3,3 / 3,3,5

2 I2(p) Dp 2 [p]

p p 2p p

Symmetry groups of regular polytopes[edit] All symmetry groups of regular polytopes are finite Coxeter
Coxeter
groups. 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 Sn+1, also known as the Coxeter
Coxeter
group of type An. The symmetry group of the n-cube and its dual, the n-cross-polytope, is Bn, and is known as the hyperoctahedral group. The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter
Coxeter
groups. In two dimensions, the dihedral groups, which are the symmetry groups of regular polygons, form the series I2(p). 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 special 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
Coxeter
groups of type Dn, E6, E7, and E8 are the symmetry groups of certain semiregular polytopes.

Table of irreducible polytope families

Family n n-simplex n-hypercube n-orthoplex n-demicube 1k2 2k1 k21 pentagonal polytope

Group An Bn

I2(p) Dn

E6 E7 E8 F4 G2

Hn

2

Triangle

Square

p-gon (example: p=7)

Hexagon

Pentagon

3

Tetrahedron

Cube

Octahedron

Tetrahedron  

Dodecahedron

Icosahedron

4

5-cell

Tesseract

16-cell

Demitesseract

24-cell

120-cell

600-cell

5

5-simplex

5-cube

5-orthoplex

5-demicube    

6

6-simplex

6-cube

6-orthoplex

6-demicube

122

221  

7

7-simplex

7-cube

7-orthoplex

7-demicube

132

231

321  

8

8-simplex

8-cube

8-orthoplex

8-demicube

142

241

421  

9

9-simplex

9-cube

9-orthoplex

9-demicube  

10

10-simplex

10-cube

10-orthoplex

10-demicube  

Affine Coxeter
Coxeter
groups[edit]

Coxeter
Coxeter
diagrams for the Affine Coxeter
Coxeter
groups

Stiefel diagram for the

G

2

displaystyle G_ 2

root system

See also: Affine Dynkin diagram
Dynkin diagram
and Affine root system The affine Coxeter
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
Coxeter
group, and the Coxeter
Coxeter
graph is obtained from the Coxeter
Coxeter
graph of the Coxeter
Coxeter
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 An in this way, and the corresponding Coxeter group is the affine Weyl group
Weyl group
of An. 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.[3] The Stiefel diagram divides the plane into infinitely many connected components called alcoves, and the affine Coxeter
Coxeter
group acts freely and transitively on the alcoves, just as the ordinary Weyl group
Weyl group
acts freely and transitively on the Weyl chambers. The figure at right illustrates the Stiefel diagram for the

G

2

displaystyle G_ 2

root system. Suppose

R

displaystyle R

is an irreducible root system of rank

r > 1

displaystyle r>1

and let

α

1

, … ,

α

r

displaystyle alpha _ 1 ,ldots ,alpha _ r

be a collection of simple roots. Let, also,

α

r + 1

displaystyle alpha _ r+1

denote the highest root. Then the affine Coxeter
Coxeter
group is generated by the ordinary (linear) reflections about the hyperplanes perpendicular to

α

1

, … ,

α

r

displaystyle alpha _ 1 ,ldots ,alpha _ r

, together with an affine reflection about a translate of the hyperplane perpendicular to

α

r + 1

displaystyle alpha _ r+1

. The Coxeter
Coxeter
graph for the affine Weyl group
Weyl group
is the Coxeter–Dynkin diagram for

R

displaystyle R

, together with one additional node associated to

α

r + 1

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

α

r + 1

displaystyle alpha _ r+1

.[4] A list of the affine Coxeter
Coxeter
groups follows:

Group symbol Witt symbol Bracket notation Coxeter graph Related uniform tessellation(s)

A ~

n

displaystyle tilde A _ n

Pn+1 [3[n]] ... or ... Simplectic honeycomb

B ~

n

displaystyle tilde B _ n

Sn+1 [4,3n-3,31,1] ... Demihypercubic honeycomb

C ~

n

displaystyle tilde C _ n

Rn+1 [4,3n-2,4] ... Hypercubic honeycomb

D ~

n

displaystyle tilde D _ n

Qn+1 [ 31,1,3n-4,31,1] ... Demihypercubic honeycomb

E ~

6

displaystyle tilde E _ 6

T7 [32,2,2] or 222

E ~

7

displaystyle tilde E _ 7

T8 [33,3,1] or 331, 133

E ~

8

displaystyle tilde E _ 8

T9 [35,2,1]

521, 251, 152

F ~

4

displaystyle tilde F _ 4

U5 [3,4,3,3]

16-cell
16-cell
honeycomb 24-cell
24-cell
honeycomb

G ~

2

displaystyle tilde G _ 2

V3 [6,3]

Hexagonal tiling
Hexagonal tiling
and Triangular tiling

I ~

1

displaystyle tilde I _ 1

W2 [∞]

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
Coxeter
groups[edit] There are infinitely many hyperbolic Coxeter
Coxeter
groups describing reflection groups in hyperbolic space, notably including the hyperbolic triangle groups. Partial orders[edit] A choice of reflection generators gives rise to a length function l on a Coxeter
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

v → ( − 1

)

l ( v )

displaystyle vto (-1)^ l(v)

defines a map

G →

± 1

,

displaystyle Gto 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
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
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[edit] Since a Coxeter
Coxeter
group

W

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

Z

2

displaystyle mathbf Z _ 2

. This may be restated in terms of the first homology group of

W

displaystyle W

. The Schur multiplier
Schur multiplier

M ( W )

displaystyle M(W)

, equal to the second homology group of

W

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

W

n

displaystyle W_ n

of finite or affine Weyl groups, the rank of

M (

W

n

)

displaystyle M(W_ n )

stabilizes as

n

displaystyle n

goes to infinity. See also[edit]

Artin group Triangle group Coxeter
Coxeter
element Coxeter
Coxeter
number Complex reflection group Chevalley–Shephard–Todd theorem Coxeter–Dynkin diagram Iwahori–Hecke algebra, a quantum deformation of the group algebra Kazhdan–Lusztig polynomial Longest element of a Coxeter
Coxeter
group Supersolvable arrangement

References[edit]

^ Brink, Brigitte; Howlett, RobertB. (1993), "A finiteness property and an automatic structure for Coxeter
Coxeter
groups", Mathematische Annalen, 296 (1): 179–190, doi:10.1007/BF01445101, Zbl 0793.20036.  ^ Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61 ^ Hall 2015 Section 13.6 ^ Hall 2015 Chapter 13, Exercises 12 and 13

Further reading[edit]

Björner, Anders; Brenti, Francesco (2005), Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231, Springer, ISBN 978-3-540-27596-1, Zbl 1110.05001  Bourbaki, Nicolas (2002), Lie Groups and Lie Algebras: Chapters 4-6, Elements of Mathematics, Springer, ISBN 978-3-540-42650-9, Zbl 0983.17001  Coxeter, H. S. M. (1934), "Discrete groups generated by reflections", Annals of Mathematics, 35 (3): 588–621, doi:10.2307/1968753, JSTOR 1968753  Coxeter, H. S. M. (1935), "The complete enumeration of finite groups of the form

r

i

2

= (

r

i

r

j

)

k

i j

= 1

displaystyle r_ i ^ 2 =(r_ i r_ j )^ k_ ij =1

", J. London Math. Soc., 1, 10 (1): 21–25, doi:10.1112/jlms/s1-10.37.21  Davis, Michael W. (2007), The Geometry and Topology of Coxeter
Coxeter
Groups (PDF), ISBN 978-0-691-13138-2, Zbl 1142.20020  Grove, Larry C.; Benson, Clark T. (1985), Finite Reflection Groups, Graduate texts in mathematics, 99, Springer, ISBN 978-0-387-96082-1  Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666  Humphreys, James E. (1992) [1990], Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, ISBN 978-0-521-43613-7, Zbl 0725.20028  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
Coxeter
groups, Research Notes in Mathematics, 54, Pitman, ISBN 978-0-273-08517-1, Zbl 0483.57002 

Ihara, S.; Yokonuma, Takeo (1965), "On the second cohomology groups (Schur-multipliers) of finite reflection groups" (PDF), Jour. 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 

Vinberg, Ernest B. (1984), "Absence of crystallographic groups of reflections in Lobachevski spaces of large dimension", Trudy Moskov. Mat. Obshch., 47  Yokonuma, Takeo (1965), "On the second cohomology groups (Schur-multipliers) of infinite discrete reflection groups", Jour. Fac. Sci. Univ. Tokyo, Sect. 1, 11: 173–186, Zbl 0136.28803 

External links[edit]

Hazewinkel, Michiel, ed. (2001) [1994], " Coxeter
Coxeter
group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4  Weisstein, Eric W. " Coxeter
Coxeter
group". MathWorld.  Jenn software for visualizing the Cayley graphs of finite Coxeter groups on up to fou

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