In mathematics , a COXETER GROUP, named after
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
Standard references include (Humphreys 1992 ) and (Davis 2007 ). CONTENTS * 1 Definition * 1.1 Coxeter matrix and
* 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
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}
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
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
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
Historically, (Coxeter 1934 ) proved that every reflection group is a
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:
Many, but not all of these, are Weyl groups, and every
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
Note further that the (directed) Dynkin diagrams Bn and Cn give rise
to the same
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
G2 - 2 12 hexagon H2 G2 2 10 pentagon H3 G3 3 120 icosahedron / dodecahedron H4 G4 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
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
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 p-gon (example: p=7 ) 3 4 5 6 122 221 7 132 231 321 8 142 241 421 9 10 AFFINE COXETER GROUPS Coxeter diagrams for the Affine Coxeter groups See also:
Affine
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
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
]
B N {DISPLAYSTYLE {TILDE {B}}_{N}} Sn+1 C N {DISPLAYSTYLE {TILDE {C}}_{N}} Rn+1
D N {DISPLAYSTYLE {TILDE {D}}_{N}} Qn+1 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
G 2 {DISPLAYSTYLE {TILDE {G}}_{2}} V3
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
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
For example, the permutation (1 2 3) in S3 has only one reduced word,
(12)(23), so covers (12) and (23) in the
HOMOLOGY Since a
The
* ^ 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,
* 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. |