In mathematics, a
Coxeter
Contents 1 Definition 1.1
Coxeter
2 An example
3 Connection with reflection groups
4 Finite
Coxeter
4.1 Classification 4.2 Weyl groups 4.3 Properties 4.4 Symmetry groups of regular polytopes 5 Affine
Coxeter
Definition[edit]
Formally, a
Coxeter
⟨ 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
S = r 1 , … , r n displaystyle S= r_ 1 ,dots ,r_ n is called a
Coxeter
S displaystyle S is not uniquely determined by W displaystyle W . For example, the
Coxeter
B 3 displaystyle B_ 3 and A 1 × A 3 displaystyle A_ 1 times A_ 3 are isomorphic but the
Coxeter
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
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
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
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 ( 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
Examples
Coxeter
I ~ 1 displaystyle tilde I _ 1 A3 B3 D4 A ~ 3 displaystyle tilde A _ 3
Coxeter
Coxeter
[ 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 nonconsecutive
transpositions always commute, while (k k+1) (k+1 k+2) gives the
3cycle (k k+2 k+1). Of course this only shows that Sn+1 is a quotient
group of the
Coxeter
( 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
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
Classification[edit]
The finite
Coxeter
A n , B n , D n , displaystyle A_ n ,B_ n ,D_ n , one oneparameter 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
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
G 2 . displaystyle G_ 2 . The nonWeyl 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
H 3 , displaystyle H_ 3 , the dodecahedron (dually, icosahedron) does not fill space; for H 4 , displaystyle H_ 4 , the
120cell
I 2 ( p ) displaystyle I_ 2 (p) a pgon 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
Rank
n
Group
symbol
Alternate
symbol
Bracket
notation
Coxeter
graph
Reflections
m=nh/2[2]
Coxeter
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 [3n1] ... n(n+1)/2 n+1 (n + 1)! nsimplex 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 [4,3n2] ... n2 2n 2n n! ncube / northoplex 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 [3n3,1,1] ... n(n1) 2(n1) 2n−1 n! ndemicube / northoplex 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
Table of irreducible polytope families Family n nsimplex nhypercube northoplex ndemicube 1k2 2k1 k21 pentagonal polytope Group An Bn I2(p) Dn E6 E7 E8 F4 G2 Hn 2 Triangle Square pgon (example: p=7) Hexagon Pentagon 3 Tetrahedron Cube Octahedron Tetrahedron Dodecahedron Icosahedron 4 5cell Tesseract 16cell Demitesseract 24cell 120cell 600cell 5 5simplex 5cube 5orthoplex 5demicube 6 6simplex 6cube 6orthoplex 6demicube 122 221 7 7simplex 7cube 7orthoplex 7demicube 132 231 321 8 8simplex 8cube 8orthoplex 8demicube 142 241 421 9 9simplex 9cube 9orthoplex 9demicube 10 10simplex 10cube 10orthoplex 10demicube Affine
Coxeter
Coxeter
Stiefel diagram for the G 2 displaystyle G_ 2 root system See also: Affine
Dynkin diagram
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
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
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
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 [4,3n3,31,1] ... Demihypercubic honeycomb C ~ n displaystyle tilde C _ n Rn+1 [4,3n2,4] ... Hypercubic honeycomb D ~ n displaystyle tilde D _ n Qn+1 [ 31,1,3n4,31,1] ... 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]
16cell
G ~ 2 displaystyle tilde G _ 2 V3 [6,3]
Hexagonal 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
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
W displaystyle W is generated by finitely many elements of order 2, its abelianization is an elementary abelian 2group, 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
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 2group. 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
References[edit] ^ Brink, Brigitte; Howlett, RobertB. (1993), "A finiteness property
and an automatic structure for
Coxeter
Further reading[edit] Björner, Anders; Brenti, Francesco (2005), Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231, Springer, ISBN 9783540275961, Zbl 1110.05001 Bourbaki, Nicolas (2002), Lie Groups and Lie Algebras: Chapters 46, Elements of Mathematics, Springer, ISBN 9783540426509, 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/s110.37.21
Davis, Michael W. (2007), The Geometry and Topology of
Coxeter
Ihara, S.; Yokonuma, Takeo (1965), "On the second cohomology groups (Schurmultipliers) of finite reflection groups" (PDF), Jour. Fac. Sci. Univ. Tokyo, Sect. 1, 11: 155–171, Zbl 0136.28802, archived from the original (PDF) on 20131023 Howlett, Robert B. (1988), "On the Schur Multipliers of Coxeter Groups", J. London Math. Soc., 2, 38 (2): 263–276, doi:10.1112/jlms/s238.2.263, 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 (Schurmultipliers) 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
