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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 ; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced (Coxeter 1934 ) as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 (Coxeter 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 .

Standard references include (Humphreys 1992 ) and (Davis 2007 ).

CONTENTS

* 1 Definition

* 1.1 Coxeter matrix and Schläfli matrix

* 2 An example * 3 Connection with reflection groups

* 4 Finite Coxeter groups

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

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

DEFINITION

Formally, a 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 group
Coxeter group
with generators S = { r 1 , , r n } {displaystyle S={r_{1},dots ,r_{n}}} is called a COXETER SYSTEM. Note that in general S {displaystyle S} is not uniquely determined by W {displaystyle W} . For example, the 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 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 MATRIX AND SCHLäFLI MATRIX

The 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 group.

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 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 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, M i j {displaystyle M_{ij}} , is related to the n n {displaystyle ntimes n} Schläfli matrix C {displaystyle C} with entries C i j = 2 cos ( / 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 is useful because its eigenvalues determine whether the Coxeter group
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 I 1 {DISPLAYSTYLE {TILDE {I}}_{1}} A3 B3 D4 A 3 {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 {displaystyle left {displaystyle left {displaystyle left {displaystyle left

For more details on this topic, see Reflection group .

Coxeter groups are deeply connected with reflection groups . Simply put, 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 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 group
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 / k {displaystyle pi /k} , with r i r j {displaystyle r_{i}r_{j}} being of order k abstracting from a rotation by 2 / k {displaystyle 2pi /k} ).

The abstract group of a reflection group is a Coxeter group, while conversely a reflection group can be seen as a linear representation of a Coxeter group. For finite reflection groups, this yields an exact correspondence: every finite Coxeter group
Coxeter group
admits a faithful representation as a finite reflection group of some Euclidean space. For infinite Coxeter groups, however, a Coxeter group
Coxeter group
may not admit a representation as a reflection group.

Historically, (Coxeter 1934 ) proved that every reflection group is a Coxeter group
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 group, while (Coxeter 1935 ) proved that every finite Coxeter group
Coxeter group
had a representation as a reflection group, and classified finite Coxeter groups.

FINITE COXETER GROUPS

Coxeter graphs of the finite Coxeter groups.

CLASSIFICATION

The finite Coxeter groups were classified in (Coxeter 1935 ), in terms of Coxeter–Dynkin diagrams ; they are all represented by reflection groups of finite-dimensional Euclidean spaces.

The finite 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

Main article: Weyl group

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 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 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 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
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 H 3 , {displaystyle H_{3},} the dodecahedron (dually, icosahedron) does not fill space; for H 4 , {displaystyle H_{4},} the 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 (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 Coxeter groups are given in the following table:

Group symbol Alternate symbol BRACKET NOTATION RANK ORDER RELATED POLYTOPES COXETER-DYNKIN DIAGRAM

AN An

n (n + 1)! n-simplex ..

BN Cn

n 2n n! n-hypercube / n-cross-polytope ...

DN Bn

n 2n−1 n! n-demihypercube ...

E6 E6

6 72x6! = 51840 221 , 122 or

E7 E7

7 72x8! = 2903040 321 , 231 , 132

E8 E8

8 192x10! = 696729600 421 , 241 , 142

F4 F4

4 1152 24-cell

G2 -

2 12 hexagon

H2 G2

2 10 pentagon

H3 G3

3 120 icosahedron / dodecahedron

H4 G4

4 14400 120-cell / 600-cell

I2(P) D2p

2 2p p-gon

SYMMETRY GROUPS OF REGULAR POLYTOPES

All symmetry groups of regular polytopes are finite 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 group
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 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 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 BCN

I2(p) Dn

E6 E7 E8 F4 G2

HN

2

Triangle

Square
Square

p-gon (example: p=7 )

Hexagon
Hexagon

Pentagon
Pentagon

3

Tetrahedron
Tetrahedron

Cube
Cube

Octahedron
Octahedron

Tetrahedron
Tetrahedron

Dodecahedron
Dodecahedron

Icosahedron
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 GROUPS

Coxeter diagrams for the Affine Coxeter groups See also: Affine Dynkin diagram and Affine root system

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 is obtained from the Coxeter graph of the 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
Coxeter group
is the affine Weyl group of An. For n = 2, this can be pictured as 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
Coxeter group
is then the group generated by the (affine) reflections about all the hyperplanes in the diagram.

A list of the affine Coxeter groups follows:

Group symbol Witt symbol BRACKET NOTATION RELATED UNIFORM TESSELLATION(S) COXETER-DYNKIN DIAGRAM

A N {DISPLAYSTYLE {TILDE {A}}_{N}} Pn+1 ] Simplectic honeycomb ... or ...

B N {DISPLAYSTYLE {TILDE {B}}_{N}} Sn+1

Demihypercubic honeycomb ...

C N {DISPLAYSTYLE {TILDE {C}}_{N}} Rn+1

Hypercubic honeycomb
Hypercubic honeycomb
...

D N {DISPLAYSTYLE {TILDE {D}}_{N}} Qn+1

Demihypercubic honeycomb ...

E 6 {DISPLAYSTYLE {TILDE {E}}_{6}} T7

222 or

E 7 {DISPLAYSTYLE {TILDE {E}}_{7}} T8

331 , 133 or

E 8 {DISPLAYSTYLE {TILDE {E}}_{8}} T9

521 , 251 , 152

F 4 {DISPLAYSTYLE {TILDE {F}}_{4}} U5

16-cell honeycomb 24-cell honeycomb

G 2 {DISPLAYSTYLE {TILDE {G}}_{2}} V3

Hexagonal tiling
Hexagonal tiling
and Triangular tiling
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 GROUPS

There are infinitely many hyperbolic Coxeter groups describing reflection groups in hyperbolic space , notably including the hyperbolic triangle groups .

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 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 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
Coxeter group
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 Z2. This may be restated in terms of the first homology group of W.

The Schur multiplier M(W) (related to the second homology) was computed in (Ihara ">

* ^ Brink, Brigitte; Howlett, RobertB. (1993), "A finiteness property and an automatic structure for Coxeter groups", Mathematische Annalen, 296 (1): 179–190, Zbl 0793.20036 , doi :10.1007/BF01445101 . * ^ Hall 2015 Section 13.6

FURTHER READING

* Coxeter, H. S. M. (1934), "Discrete groups generated by reflections", Ann. of Math., 35 (3): 588–621, JSTOR
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 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 * Humphreys, James E. (1992) , 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 * Björner, Anders ; Brenti, Francesco (2005), Combinatorics of Coxeter Groups, Graduate Texts in Mathematics
Mathematics
, 231, Springer, ISBN 978-3-540-27596-1 , Zbl 1110.05001 * Hiller, Howard (1982), Geometry of Coxeter groups, Research Notes in Mathematics, 54, Pitman, ISBN 978-0-273-08517-1 , Zbl 0483.57002 * Bourbaki, Nicolas (2002), Lie Groups and Lie Algebras: Chapters 4-6, Elements of Mathematics, Springer, ISBN 978-3-540-42650-9 , Zbl 0983.17001 * Howlett, Robert B. (1988), "On the Schur Multipliers of Coxeter Groups", J. London Math. Soc., 2, 38 (2): 263–276, Zbl 0627.20019 , doi :10.1112/jlms/s2-38.2.263

* Vinberg, E. B. (1984), "Absence of crystallographic groups of reflections in Lobachevski spaces of large dimension", Trudy Moskov. Mat. Obshch., 47 * 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 * 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

* Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer

EXTERNAL LINKS

* Hazewinkel,

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