Coxeter groupsFrom Wikipedia, the free encyclopedia

Contents

1 Bruhat order 11.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Bruhat graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Coxeter complex 32.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 The canonical linear representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Chambers and the Tits cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.3 The Coxeter complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.1 Finite dihedral groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 The infinite dihedral group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Alternative construction of the Coxeter complex . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Coxeter element 63.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Group order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Coxeter elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Coxeter plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 H4 (mathematics) 104.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.1.1 Coxeter matrix and Schlfli matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Connection with reflection groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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ii CONTENTS

4.4 Finite Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4.2 Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4.4 Symmetry groups of regular polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.5 Affine Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 Hyperbolic Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.7 Partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.8 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Coxeter matroid 165.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Relation to matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 CoxeterDynkin diagram 176.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.2 Schlfli matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.2.1 Rank 2 Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.2.2 Geometric visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.3 Finite Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.4 Application with uniform polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.4.1 Example polyhedra and tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.5 Affine Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.6 Hyperbolic Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6.6.1 Hyperbolic groups in H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.6.2 Compact (Lannr simplex groups) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.6.3 Paracompact (Koszul simplex groups) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.6.4 Hypercompact Coxeter groups (Vinberg polytopes) . . . . . . . . . . . . . . . . . . . . . 25

6.7 Lorentzian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.7.1 Very-extended Coxeter Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.8 Geometric folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 H4 (mathematics) 31

CONTENTS iii

7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.1.1 Coxeter matrix and Schlfli matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.3 Connection with reflection groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.4 Finite Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.4.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.4.2 Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.4.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4.4 Symmetry groups of regular polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.5 Affine Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.6 Hyperbolic Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.7 Partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.8 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 Longest element of a Coxeter group 378.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9 Triangle group 399.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9.2.1 The Euclidean case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.2.2 The spherical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.2.3 The hyperbolic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9.3 von Dyck groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.4 Overlapping tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.10 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapter 1

Bruhat order

In mathematics, theBruhat order (also called strong order or strong Bruhat order orChevalley order orBruhatChevalley order or ChevalleyBruhat order) is a partial order on the elements of a Coxeter group, that correspondsto the inclusion order on Schubert varieties.

1.1 History

The Bruhat order on the Schubert varieties of a flag manifold or Grassmannian was first studied by Ehresmann(1934), and the analogue for more general semisimple algebraic groups was studied by Chevalley (1958). Verma(1968) started the combinatorial study of the Bruhat order on the Weyl group, and introduced the name Bruhatorder because of the relation to the Bruhat decomposition introduced by Franois Bruhat.The left and right weak Bruhat orderings were studied by Bjrner (1984).

1.2 Definition

If (W,S) is a Coxeter system with generators S, then the Bruhat order is a partial order on the group W. Recall thata reduced word for an element w of W is a minimal length expression of w as a product of elements of S, and thelength l(w) of w is the length of a reduced word.

The (strong) Bruhat order is defined by uv if some substring of some (or every) reduced word for v is areduced word for u.

(Note that here a substring is not necessarily a consecutive substring.)

The weak left (Bruhat) order is defined by uLv if some final substring of some reduced word for v is a reducedword for u.

The weak right (Bruhat) order is defined by uRv if some initial substring of some reduced word for v is areduced word for u.

For more on the weak orders, see the article weak order of permutations.

1.3 Bruhat graph

The Bruhat graph is a directed graph related to the (strong) Bruhat order. The vertex set is the set of elements of theCoxeter group and the edge set consists of directed edges (u, v) whenever u = t v for some reflection t and l(u) < l(v).One may view the graph as an edge-labeled directed graph with edge labels coming from the set of reflections. (One

1

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2 CHAPTER 1. BRUHAT ORDER

could also define the Bruhat graph using multiplication on the right; as graphs, the resulting objects are isomorphic,but the edge labelings are different.)The strong Bruhat order on the symmetric group (permutations) has Mbius function given by(, ) = (1)()(), and thus this poset is Eulerian, meaning its Mbius function is produced by the rank function on the poset.

1.4 References Bjrner, Anders (1984), Orderings of Coxeter groups, in Greene, Curtis, Combinatorics and algebra (Boul-der, Colo., 1983), Contemp. Math. 34, Providence, R.I.: American Mathematical Society, pp. 175195,ISBN 978-0-8218-5029-9, MR 777701

Bjrner, Anders; Brenti, Francesco (2005), Combinatorics of Coxeter groups, Graduate Texts in Mathemat-ics 231, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-27596-7, ISBN 978-3-540-44238-7, MR2133266

Chevalley, C. (1958), Sur les dcompositions cellulaires des espaces G/B, in Haboush, William J.; Parshall,Brian J., Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), Proc. Sym-pos. Pure Math. 56, Providence, R.I.: American Mathematical Society, pp. 123, ISBN 978-0-8218-1540-3,MR 1278698

Ehresmann, Charles (1934), Sur la Topologie de Certains Espaces Homognes, Annals of Mathematics,Second Series (in French) (Annals of Mathematics) 35 (2): 396443, ISSN 0003-486X, JFM 60.1223.05,JSTOR 1968440

Verma, Daya-Nand (1968), Structure of certain induced representations of complex semisimple Lie algebras,Bulletin of the American Mathematical Society 74: 160166, doi:10.1090/S0002-9904-1968-11921-4, ISSN0002-9904, MR 0218417

http://books.google.com/books?id=2axt00oBDEwC&pg=175https://en.wikipedia.org/wiki/Curtis_Greenehttp://books.google.com/books?id=2axt00oBDEwChttp://books.google.com/books?id=2axt00oBDEwChttps://en.wikipedia.org/wiki/American_Mathematical_Societyhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-5029-9https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=777701https://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252F3-540-27596-7https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-44238-7https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=2133266http://books.google.com/books?id=-fTjI8adsNQChttps://en.wikipedia.org/wiki/American_Mathematical_Societyhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-1540-3https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1278698https://en.wikipedia.org/wiki/Annals_of_Mathematicshttps://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0003-486Xhttps://en.wikipedia.org/wiki/Jahrbuch_%C3%BCber_die_Fortschritte_der_Mathematikhttps://zbmath.org/?format=complete&q=an:60.1223.05https://en.wikipedia.org/wiki/JSTORhttps://www.jstor.org/stable/1968440https://en.wikipedia.org/wiki/Bulletin_of_the_American_Mathematical_Societyhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1090%252FS0002-9904-1968-11921-4https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0002-9904https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=0218417

Chapter 2

Coxeter complex

In mathematics, theCoxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex)associated to a Coxeter group. Coxeter complexes are the basic objects that allow the construction of buildings; theyform the apartments of a building.

2.1 Construction

2.1.1 The canonical linear representation

The first ingredient in the construction of the Coxeter complex associated to a Coxeter groupW is a certain representationof W, called the canonical representation of W.Let (W,S) be a Coxeter system associated to W, with Coxeter matrix M = (m(s, t))s,tS . The canonical rep-resentation is given by a vector space V with basis of formal symbols (es)sS , which is equipped with the sym-metric bilinear form B(es, et) = cos

(

m(s,t)

). The action of W on this vector space V is then given by

s(v) = v 2 B(es,v)B(es,es)es , as motivated by the expression for reflections in root systems.This representation has several foundational properties in the theory of Coxeter groups; for instance, the bilinear formB is positive definite if and only if W is finite. It is (always) a faithful representation of W.

2.1.2 Chambers and the Tits cone

One can think of this representation as expressing W as some sort of reflection group, with the caveat that B mightnot be positive definite. It becomes important then to distinguish the representation V with its dual V*. The vectorses lie in V, and have corresponding dual vectors es in V*, given by:

es , v = 2B(es, v)

B(es, es),

where the angled brackets indicate the natural pairing of a dual vector in V* with a vector of V, and B is the bilinearform as above.Now W acts on V*, and the action satisfies the formula

s(f) = f f, eses ,

for s S and any f in V*. This expresses s as a reflection in the hyperplane Hs = {f V : f, es = 0} . Onehas the fundamental chamber C = {f V : f, es > 0 s S} , this has faces the so-called walls, Hs . Theother chambers can be obtained from C by translation: they are the wC for w W .

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4 CHAPTER 2. COXETER COMPLEX

Given a fundamental chamber C , the Tits cone is defined to be X =

wW wC . This need not be the whole of V*.Of major importance is the fact that the Tits cone X is convex. The action of W on the Tits cone X has fundamentaldomain the fundamental chamber C .

2.1.3 The Coxeter complex

Once one has defined the Tits cone X, the Coxeter complex (W,S) of W with respect to S can be defined as thequotient of X, with the origin removed, by the positive reals (+, ):

(W,S) = (X \ {0})/R+

2.2 Examples

2.2.1 Finite dihedral groups

The dihedral groups Dn (of order 2n) are Coxeter groups, of corresponding type I2(n) . These have the presentations, t

s2, t2, (st)n .The canonical linear representation of I2(n) is the usual reflection representation of the dihedral group, as acting ona n-gon in the plane (so V = R2 in this case). For instance, in the case n = 3, we get the Coxeter group of typeI2(3) = A2 , acting on an equilateral triangle in the plane. Each reflection s has an associated hyperplane Hs in thedual vector space (which can be canonically identified with the vector space itself using the bilinear form B, which isan inner product in this case as remarked above), these are the walls. They cut out chambers, as seen below:

The Coxeter complex is then the corresponding 2n-gon, as in the previous image. This is a simplicial complex ofdimension 1, and it can be colored by cotype.

2.2.2 The infinite dihedral group

Another motivating example is the infinite dihedral group D . This can be seen as the group of symmetries of thereal line that preserves the set of points with integer coordinates; it is generated by the reflections in x = 0 and x = 12. This group has the Coxeter presentation

s, t

s2, t2 .In this case, it is no longer possible to identify V with the dual space V*, as B is not positive definite. It is then betterto work solely with V*, which is where the hyperplanes are defined. This then gives the following picture:

https://en.wikipedia.org/wiki/Fundamental_domainhttps://en.wikipedia.org/wiki/Fundamental_domainhttps://en.wikipedia.org/wiki/Positive_realshttps://en.wikipedia.org/wiki/Infinite_dihedral_group

2.3. ALTERNATIVE CONSTRUCTION OF THE COXETER COMPLEX 5

In this case, the Tits cone is not the whole plane, but only the upper half plane. Quotienting out by the positive realsthen yields another copy of the real line, with marked points at the integers. This is the Coxeter complex of theinfinite dihedral group.

2.3 Alternative construction of the Coxeter complex

Another description of the Coxeter complex uses standard cosets of the Coxeter group W. A standard coset is a cosetof the form wWJ , where WJ = J for some subset J of S. For instance, WS = W and W = {1} .The Coxeter complex (W,S) is then the poset of standard cosets, ordered by reverse inclusion. This has a canonicalstructure of a simplicial complex, as do all posets that satisfy:

Any two elements have a greatest lower bound.

The poset of elements less than or equal to any given element is isomorphic to the poset of subsets of {1, 2, . . . , n}for some integer n.

2.4 Properties

The Coxeter complex associated to (W,S) has dimension |S| 1 . It is homeomorphic to a (|S| 1) -sphere if Wis finite and is contractible if W is infinite.

2.5 See also Buildings

Weyl group

Root system

2.6 References Peter Abramenko and Kenneth S. Brown, Buildings, Theory and Applications. Springer, 2008.

https://en.wikipedia.org/wiki/Posethttps://en.wikipedia.org/wiki/Contractible_spacehttps://en.wikipedia.org/wiki/Building_(mathematics)https://en.wikipedia.org/wiki/Weyl_grouphttps://en.wikipedia.org/wiki/Root_system

Chapter 3

Coxeter element

Not to be confused with Longest element of a Coxeter group.

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group, hencealso of a root system or its Weyl group. It is named after H.S.M. Coxeter.[1]

3.1 Definitions

Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classesof Coxeter elements, and they have infinite order.There are many different ways to define the Coxeter number h of an irreducible root system.A Coxeter element is a product of all simple reflections. The product depends on the order in which they are taken,but different orderings produce conjugate elements, which have the same order.

The Coxeter number is the number of roots divided by the rank. The number of mirrors in the Coxeter groupis half the number of roots.

The Coxeter number is the order of a Coxeter element; note that conjugate elements have the same order.

If the highest root is mi for simple roots i, then the Coxeter number is 1 + m

The dimension of the corresponding Lie algebra is n(h + 1), where n is the rank and h is the Coxeter number.

The Coxeter number is the highest degree of a fundamental invariant of the Weyl group acting on polynomials.

The Coxeter number is given by the following table:

The invariants of the Coxeter group acting on polynomials form a polynomial algebra whose generators are thefundamental invariants; their degrees are given in the table above. Notice that if m is a degree of a fundamentalinvariant then so is h + 2 m.The eigenvalues of a Coxeter element are the numbers e2i(m 1)/h as m runs through the degrees of the fundamentalinvariants. Since this starts with m = 2, these include the primitive hth root of unity, h = e2i/h, which is importantin the Coxeter plane, below.

3.2 Group order

There are relations between group order, g, and the Coxeter number, h:[2]

[p]: 2h/g = 1

6

https://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_grouphttps://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Order_(group_theory)https://en.wikipedia.org/wiki/Coxeter_grouphttps://en.wikipedia.org/wiki/Root_systemhttps://en.wikipedia.org/wiki/Weyl_grouphttps://en.wikipedia.org/wiki/H.S.M._Coxeterhttps://en.wikipedia.org/wiki/Conjugacy_classhttps://en.wikipedia.org/wiki/Conjugate_elementshttps://en.wikipedia.org/wiki/Order_(group_theory)https://en.wikipedia.org/wiki/Lie_algebrahttps://en.wikipedia.org/wiki/Primitive_root_of_unityhttps://en.wikipedia.org/wiki/Coxeter_element#Coxeter_plane

3.3. COXETER ELEMENTS 7

[p,q]: 8/g , = 2/p + 2/q 1

[p,q,r]: 64h/g , , = 12 - p - 2q - r + 4/p + 4/r

[p,q,r,s]: 16/g , ,, = 8/g , , + 8/g ,, + 2/(ps) - 1/p - 1/q - 1/r - 1/s +1

...

An example, [3,3,5] has h=30, so 64*30/g = 12 - 3 - 6 - 5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2= 960*15 = 14400.

3.3 Coxeter elements

Coxeter elements of An1 = Sn , considered as the symmetric group on n elements, are n-cycles: for simplereflections the adjacent transpositions (1, 2), (2, 3), . . . , a Coxeter element is the n-cycle (1, 2, 3, . . . , n) .[3]

The dihedral group Dihm is generated by two reflections that form an angle of 2/2m , and thus their product is arotation by 2/m .

3.4 Coxeter plane

For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2/h. This is called theCoxeter plane and is the plane on which P has eigenvalues e2i/h and e2i/h = e2i(h1)/h.[4] This plane was firstsystematically studied in (Coxeter 1948),[5] and subsequently used in (Steinberg 1959) to provide uniform proofsabout properties of Coxeter elements.[5]

The Coxeter plane is often used to draw diagrams of higher-dimensional polytopes and root systems the vertices andedges of the polytope, or roots (and some edges connecting these) are orthogonally projected onto the Coxeter plane,yielding a Petrie polygon with h-fold rotational symmetry.[6] For root systems, no root maps to zero, correspondingto the Coxeter element not fixing any root or rather axis (not having eigenvalue 1 or 1), so the projections of orbitsunder w form h-fold circular arrangements[6] and there is an empty center, as in the E8 diagram at above right. Forpolytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below forthe Platonic solids.In three dimensions, the symmetry of a regular polyhedron, {p,q}, with one directed petrie polygon marked, definedas a composite of 3 reflections, has rotoinversion symmetry S , [2+,h+], order h. Adding a mirror, the symmetry canbe doubled to antiprismatic symmetry, D , [2+,h], order 2h. In orthogonal 2D projection, this becomes dihedralsymmetry, Dihh, [h], order 2h.In four dimension, the symmetry of a regular polychoron, {p,q,r}, with one directed petrie polygon marked is a doublerotation, defined as a composite of 4 reflections, with symmetry +1/ [C C ][7] (John H. Conway), (C /C1;C /C1)(#1', Patrick du Val (1964)[8]), order h.In five dimension, the symmetry of a regular polyteron, {p,q,r,s}, with one directed petrie polygon marked, is repre-sented by the composite of 5 reflections.

3.5 See also

Longest element of a Coxeter group

3.6 Notes[1] Coxeter, Harold Scott Macdonald; Chandler Davis, Erlich W. Ellers (2006), The Coxeter Legacy: Reflections and Projec-

tions, AMS Bookstore, p. 112, ISBN 978-0-8218-3722-1

[2] Regular polytopes, p. 233

[3] (Humphreys 1992, p. 75)

https://en.wikipedia.org/wiki/Symmetric_grouphttps://en.wikipedia.org/wiki/Dihedral_grouphttps://en.wikipedia.org/wiki/Coxeter_element#CITEREFCoxeter1948https://en.wikipedia.org/wiki/Coxeter_element#CITEREFSteinberg1959https://en.wikipedia.org/wiki/Orthogonal_projectionhttps://en.wikipedia.org/wiki/Petrie_polygonhttps://en.wikipedia.org/wiki/Platonic_solidhttps://en.wikipedia.org/wiki/Regular_polyhedronhttps://en.wikipedia.org/wiki/Rotoinversionhttps://en.wikipedia.org/wiki/Dihedral_symmetryhttps://en.wikipedia.org/wiki/Dihedral_symmetryhttps://en.wikipedia.org/wiki/Convex_regular_polychoronhttps://en.wikipedia.org/wiki/Double_rotationhttps://en.wikipedia.org/wiki/Double_rotationhttps://en.wikipedia.org/wiki/John_H._Conwayhttps://en.wikipedia.org/wiki/Patrick_du_Valhttps://en.wikipedia.org/wiki/Regular_polyteronhttps://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_grouphttp://books.google.com/?id=cKpBGcqpspIC&pg=PA107&dq=%2522Coxeter+number%2522+%2522Donald+Coxeter%2522http://books.google.com/?id=cKpBGcqpspIC&pg=PA107&dq=%2522Coxeter+number%2522+%2522Donald+Coxeter%2522https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-3722-1https://en.wikipedia.org/wiki/Coxeter_element#CITEREFHumphreys1992http://books.google.com/books?id=ODfjmOeNLMUC&pg=PA75&dq=%2522coxeter+element%2522

8 CHAPTER 3. COXETER ELEMENT

Projection of E8 root system onto Coxeter plane, showing 30-fold symmetry.

[4] (Humphreys 1992, Section 3.17, Action on a Plane, pp. 7678)

[5] (Reading 2010, p. 2)

[6] (Stembridge 2007)

[7] On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5

[8] Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford,1964.

3.7 References Coxeter, H. S. M. (1948), Regular Polytopes, Methuen and Co.

Steinberg, R. (June 1959), Finite Reflection Groups, Transactions of the American Mathematical Society 91(3): 493504, doi:10.1090/S0002-9947-1959-0106428-2, ISSN 0002-9947, JSTOR 1993261

Hiller, Howard Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publish-ing Program), Boston, Mass.-London, 1982. iv+213 pp. ISBN 0-273-08517-4

https://en.wikipedia.org/wiki/Coxeter_element#CITEREFHumphreys1992http://books.google.com/books?id=ODfjmOeNLMUC&pg=PA76https://en.wikipedia.org/wiki/Coxeter_element#CITEREFReading2010https://en.wikipedia.org/wiki/Coxeter_element#CITEREFStembridge2007http://www.amazon.com/Quaternions-Octonions-John-Horton-Conway/dp/1568811349https://en.wikipedia.org/wiki/Special:BookSources/9781568811345https://en.wikipedia.org/wiki/Oxford_University_Presshttps://en.wikipedia.org/wiki/Oxfordhttps://en.wikipedia.org/wiki/H._S._M._Coxeterhttps://en.wikipedia.org/wiki/Regular_Polytopes_(book)https://en.wikipedia.org/wiki/Transactions_of_the_American_Mathematical_Societyhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1090%252FS0002-9947-1959-0106428-2https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0002-9947https://en.wikipedia.org/wiki/JSTORhttps://www.jstor.org/stable/1993261https://en.wikipedia.org/wiki/Special:BookSources/0273085174

3.7. REFERENCES 9

Humphreys, James E. (1992), Reflection Groups and Coxeter Groups, Cambridge University Press, pp. 7476(Section 3.16, Coxeter Elements), ISBN 978-0-521-43613-7

Stembridge, John (April 9, 2007), Coxeter Planes

Stekolshchik, R. (2008), Notes on Coxeter Transformations and the McKay Correspondence, Springer Mono-graphs in Mathematics, doi:10.1007/978-3-540-77398-3, ISBN 978-3-540-77398-6

Reading, Nathan (2010), Noncrossing Partitions, Clusters and the Coxeter Plane, Sminaire Lotharingien deCombinatoire B63b: 32

http://books.google.com/?id=ODfjmOeNLMUChttps://en.wikipedia.org/wiki/Cambridge_University_Presshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-43613-7http://www.math.lsa.umich.edu/~jrs/coxplane.htmlhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252F978-3-540-77398-3https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-77398-6http://www.emis.de/journals/SLC/wpapers/s63reading.htmlhttps://en.wikipedia.org/wiki/S%C3%A9minaire_Lotharingien_de_Combinatoirehttps://en.wikipedia.org/wiki/S%C3%A9minaire_Lotharingien_de_Combinatoire

Chapter 4

H4 (mathematics)

In mathematics, aCoxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal descriptionin terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclideanreflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups arefinite, 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 thesymmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxetergroups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolicplane, and the Weyl groups of infinite-dimensional KacMoody algebras.Standard references include (Humphreys 1990) and (Davis 2007).

4.1 Definition

Formally, a Coxeter group can be defined as a group with the presentation

r1, r2, . . . , rn | (rirj)mij = 1

where mii = 1 and mij 2 for i = j . The condition mij = means no relation of the form (rirj)m should beimposed.The pair (W,S) where W is a Coxeter group with generators S={r1,...,rn} is called Coxeter system. Note that ingeneral S is not uniquely determined by W. For example, the Coxeter groups of type BC3 and A1xA3 are isomorphicbut 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 mi i = 1 means that (riri )1 = (ri )2 = 1 for all i ; the generators are involutions.

If mi j = 2, then the generators ri and rj commute. This follows by observing that

xx = yy = 1,together with

xyxy = 1

implies that

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

Alternatively, since the generators are involutions, ri = r1i , so (rirj)2 = rirjrirj = rirjr1i r1j ,and thus is equal to the commutator.

10

https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Harold_Scott_MacDonald_Coxeterhttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Group_presentationhttps://en.wikipedia.org/wiki/Kaleidoscopehttps://en.wikipedia.org/wiki/Reflection_grouphttps://en.wikipedia.org/wiki/Reflection_grouphttps://en.wikipedia.org/wiki/Symmetry_grouphttps://en.wikipedia.org/wiki/Regular_polyhedronhttps://en.wikipedia.org/wiki/Coxeter_group#CITEREFCoxeter1934https://en.wikipedia.org/wiki/Reflection_grouphttps://en.wikipedia.org/wiki/Coxeter_group#CITEREFCoxeter1935https://en.wikipedia.org/wiki/Symmetry_grouphttps://en.wikipedia.org/wiki/Regular_polytopehttps://en.wikipedia.org/wiki/Weyl_grouphttps://en.wikipedia.org/wiki/Simple_Lie_algebrahttps://en.wikipedia.org/wiki/Triangle_grouphttps://en.wikipedia.org/wiki/Tessellation#Regular_and_irregular_tessellationshttps://en.wikipedia.org/wiki/Euclidean_planehttps://en.wikipedia.org/wiki/Hyperbolic_spacehttps://en.wikipedia.org/wiki/Hyperbolic_spacehttps://en.wikipedia.org/wiki/Kac%E2%80%93Moody_algebrahttps://en.wikipedia.org/wiki/Coxeter_group#CITEREFHumphreys1990https://en.wikipedia.org/wiki/Coxeter_group#CITEREFDavis2007https://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Presentation_of_a_grouphttps://en.wikipedia.org/wiki/Involution_(mathematics)https://en.wikipedia.org/wiki/Commutator

4.2. AN EXAMPLE 11

In order to avoid redundancy among the relations, it is necessary to assume that mi j = mj i. This follows byobserving that

yy = 1,together with

(xy)m = 1

implies that

(yx)m = (yx)myy = y(xy)my = yy = 1.

Alternatively, (xy)k and (yx)k are conjugate elements, as y(xy)ky1 = (yx)kyy1 = (yx)k .

4.1.1 Coxeter matrix and Schlfli matrix

The Coxeter matrix is the nn, symmetric matrix with entries mi j. Indeed, every symmetric matrix with positiveinteger and entries and with 1s on the diagonal such that all nondiagonal entries are greater than 1 serves to definea 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 and j are connected if and only if mi j 3.

An edge is labelled with the value of mi j whenever it is 4 or greater.

In particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxetergraph has two or more connected components, the associated group is the direct product of the groups associated tothe individual components. Thus the disjoint union of Coxeter graphs yields a direct product of Coxeter groups.The Coxeter matrix, M, , is related to the Schlfli matrix, C, , but the elements are modified, being proportional tothe dot product of the pairwise generators: Schlfli matrix C, =2cos(/M, ). The Schlfli matrix is useful becauseits eigenvalues determine whether the Coxeter group is of finite type (all positive), affine type (all non-negative, at leastone zero), or indefinite type (otherwise). The indefinite type is sometimes further subdivided, e.g. into hyperbolicand other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups.

4.2 An example

The graph in which vertices 1 through n are placed in a row with each vertex connected by an unlabelled edge to itsimmediate neighbors gives rise to the symmetric group Sn; the generators correspond to the transpositions (1 2), (23), ... (n n+1). Two non-consecutive transpositions always commute, while (k k+1) (k+1 k+2) gives the 3-cycle (kk+2 k+1). Of course this only shows that Sn+1 is a quotient group of the Coxeter group described by the graph, butit is not too difficult to check that equality holds.

4.3 Connection with reflection groups

For more details on this topic, see Reflection group.

Coxeter groups are deeply connected with reflection groups. Simply put, Coxeter groups are abstract groups (givenvia a presentation), while reflection groups are concrete groups (given as subgroups of linear groups or various gen-eralizations). Coxeter groups grew out of the study of reflection groups they are an abstraction: a reflection groupis a subgroup of a linear group generated by reflections (which have order 2), while a Coxeter group is an abstractgroup generated by involutions (elements of order 2, abstracting from reflections), and whose relations have a certainform ( (rirj)k , corresponding to hyperplanes meeting at an angle of /k , with rirj being of order k abstractingfrom a rotation by 2/k ).

https://en.wikipedia.org/wiki/Conjugate_elementshttps://en.wikipedia.org/wiki/Symmetric_matrixhttps://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttps://en.wikipedia.org/wiki/Commutative_operationhttps://en.wikipedia.org/wiki/Connected_component_(graph_theory)https://en.wikipedia.org/wiki/Direct_product_of_groupshttps://en.wikipedia.org/wiki/Disjoint_unionhttps://en.wikipedia.org/wiki/Direct_product_of_groupshttps://en.wikipedia.org/wiki/Schl%C3%A4fli_matrixhttps://en.wikipedia.org/wiki/Dot_producthttps://en.wikipedia.org/wiki/Eigenvalueshttps://en.wikipedia.org/wiki/Vertex_(graph_theory)https://en.wikipedia.org/wiki/Edge_(graph_theory)https://en.wikipedia.org/wiki/Symmetric_grouphttps://en.wikipedia.org/wiki/Generating_set_of_a_grouphttps://en.wikipedia.org/wiki/Transposition_(mathematics)https://en.wikipedia.org/wiki/Quotient_grouphttps://en.wikipedia.org/wiki/Reflection_grouphttps://en.wikipedia.org/wiki/Reflection_grouphttps://en.wikipedia.org/wiki/Linear_group

12 CHAPTER 4. H4 (MATHEMATICS)

The abstract group of a reflection group is a Coxeter group, while conversely a reflection group can be seen as alinear representation of a Coxeter group. For finite reflection groups, this yields an exact correspondence: everyfinite Coxeter group admits a faithful representation as a finite reflection group of some Euclidean space. For infiniteCoxeter groups, however, a Coxeter group may not admit a representation as a reflection group.Historically, (Coxeter 1934) proved that every reflection group is a Coxeter group (i.e., has a presentation whereall relations are of the form r2i or (rirj)k ), and indeed this paper introduced the notion of a Coxeter group, while(Coxeter 1935) proved that every finite Coxeter group had a representation as a reflection group, and classified finiteCoxeter groups.

4.4 Finite Coxeter groups

Coxeter graphs of the finite Coxeter groups.

4.4.1 Classification

The finite Coxeter groups were classified in (Coxeter 1935), in terms of CoxeterDynkin diagrams; they are allrepresented by reflection groups of finite-dimensional Euclidean spaces.The finite Coxeter groups consist of three one-parameter families of increasing rankAn, BCn, Dn, one one-parameterfamily of dimension two, I2(p), and six exceptional groups: E6, E7, E8, F4,H3, and H4.

4.4.2 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 Weylgroups are the families An, BCn, and Dn, and the exceptions E6, E7, E8, F4, and I2(6), denoted in Weyl groupnotation as G2. The non-Weyl groups are the exceptions H3 and H4, and the family I2(p) except where this coincideswith one of the Weyl groups (namely I2(3) = A2, I2(4) = BC2, and I2(6) = G2 ).This can be proven by comparing the restrictions on (undirected) Dynkin diagrams with the restrictions on Coxeterdiagrams of finite groups: formally, the Coxeter graph can be obtained from the Dynkin diagram by discarding thedirection of the edges, and replacing every double edge with an edge labelled 4 and every triple edge by an edgelabelled 6. Also note that every finitely generated Coxeter group is an Automatic group.[1] Dynkin diagrams have theadditional restriction that the only permitted edge labels are 2, 3, 4, and 6, which yields the above. Geometrically, thiscorresponds to the crystallographic restriction theorem, and the fact that excluded polytopes do not fill space or tilethe plane for H3, the dodecahedron (dually, icosahedron) does not fill space; for H4, the 120-cell (dually, 600-cell)does not fill space; for I2(p) a p-gon does not tile the plane except for p = 3, 4, or 6 (the triangular, square, andhexagonal 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 forthe Weyl group; this corresponds to the hypercube and cross-polytope being different regular polytopes but havingthe same symmetry group.

https://en.wikipedia.org/wiki/Linear_representationhttps://en.wikipedia.org/wiki/Coxeter_group#CITEREFCoxeter1934https://en.wikipedia.org/wiki/Coxeter_group#CITEREFCoxeter1935https://en.wikipedia.org/wiki/Coxeter_group#CITEREFCoxeter1935https://en.wikipedia.org/wiki/Coxeter%E2%80%93Dynkin_diagramhttps://en.wikipedia.org/wiki/Reflection_grouphttps://en.wikipedia.org/wiki/Exceptional_objecthttps://en.wikipedia.org/wiki/Weyl_grouphttps://en.wikipedia.org/wiki/Weyl_grouphttps://en.wikipedia.org/wiki/Dynkin_diagramhttps://en.wikipedia.org/wiki/Coxeter%E2%80%93Dynkin_diagramhttps://en.wikipedia.org/wiki/Dynkin_diagramhttps://en.wikipedia.org/wiki/Automatic_grouphttps://en.wikipedia.org/wiki/Crystallographic_restriction_theoremhttps://en.wikipedia.org/wiki/Hypercubehttps://en.wikipedia.org/wiki/Cross-polytope

4.5. AFFINE COXETER GROUPS 13

4.4.3 Properties

Some properties of the finite Coxeter groups are given in the following table:

4.4.4 Symmetry groups of regular polytopes

All symmetry groups of regular polytopes are finite Coxeter groups. Note that dual polytopes have the same symmetrygroup.There are three series of regular polytopes in all dimensions. The symmetry group of a regular n-simplex is thesymmetric group Sn, also known as the Coxeter group of type An. The symmetry group of the n-cube and its dual,the n-cross-polytope, is BCn, and is known as the hyperoctahedral group.The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter groups. In twodimensions, the dihedral groups, which are the symmetry groups of regular polygons, form the series I2(p). In threedimensions, the symmetry group of the regular dodecahedron and its dual, the regular icosahedron, is H3, known asthe full icosahedral group. In four dimensions, there are three special regular polytopes, the 24-cell, the 120-cell, andthe 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.

4.5 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, buteach contains a normal abelian subgroup such that the corresponding quotient group is finite. In each case, the quotientgroup is itself a Coxeter group, and the Coxeter graph is obtained from the Coxeter graph of the Coxeter group byadding another vertex and one or two additional edges. For example, for n 2, the graph consisting of n+1 verticesin a circle is obtained from An in this way, and the corresponding Coxeter group is the affine Weyl group of An. Forn = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles.A list of the affine Coxeter groups follows:The subscript is one less than the number of nodes in each case, since each of these groups was obtained by addinga node to a finite groups graph.

https://en.wikipedia.org/wiki/Symmetry_grouphttps://en.wikipedia.org/wiki/Regular_polytopehttps://en.wikipedia.org/wiki/Dual_polytopehttps://en.wikipedia.org/wiki/Simplexhttps://en.wikipedia.org/wiki/Symmetric_grouphttps://en.wikipedia.org/wiki/Cubehttps://en.wikipedia.org/wiki/Cross-polytopehttps://en.wikipedia.org/wiki/Hyperoctahedral_grouphttps://en.wikipedia.org/wiki/Dihedral_grouphttps://en.wikipedia.org/wiki/Regular_polygonhttps://en.wikipedia.org/wiki/Dodecahedronhttps://en.wikipedia.org/wiki/Icosahedronhttps://en.wikipedia.org/wiki/Full_icosahedral_grouphttps://en.wikipedia.org/wiki/24-cellhttps://en.wikipedia.org/wiki/120-cellhttps://en.wikipedia.org/wiki/600-cellhttps://en.wikipedia.org/wiki/Thorold_Gossethttps://en.wikipedia.org/wiki/Affine_Dynkin_diagramhttps://en.wikipedia.org/wiki/Affine_root_systemhttps://en.wikipedia.org/wiki/Normal_subgrouphttps://en.wikipedia.org/wiki/Abelian_grouphttps://en.wikipedia.org/wiki/Subgrouphttps://en.wikipedia.org/wiki/Quotient_group

14 CHAPTER 4. H4 (MATHEMATICS)

4.6 Hyperbolic Coxeter groups

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

4.7 Partial orders

A choice of reflection generators gives rise to a length function l on a Coxeter group, namely the minimum numberof uses of generators required to express a group element; this is precisely the length in the word metric in the Cayleygraph. An expression for v using l(v) generators is a reduced word. For example, the permutation (13) in S3 has tworeduced words, (12)(23)(12) and (23)(12)(23). The function v (1)l(v) defines a map G {1}, generalizingthe sign map for the symmetric group.Using reduced words one may define three partial orders on the Coxeter group, the (right) weak order, the absoluteorder and the Bruhat order (named for Franois Bruhat). An element v exceeds an element u in the Bruhat order ifsome (or equivalently, any) reduced word for v contains a reduced word for u as a substring, where some letters (inany position) are dropped. In the weak order, v u if some reduced word for v contains a reduced word for u as aninitial segment. Indeed, the word length makes this into a graded poset. The Hasse diagrams corresponding to theseorders are objects of study, and are related to the Cayley graph determined by the generators. The absolute order isdefined analogously to the weak order, but with generating set/alphabet consisting of all conjugates of the Coxetergenerators.For example, the permutation (1 2 3) in S3 has only one reduced word, (12)(23), so covers (12) and (23) in the Bruhatorder but only covers (12) in the weak order.

4.8 Homology

Since a Coxeter group W is generated by finitely many elements of order 2, its abelianization is an elementary abelian2-group, i.e. it is isomorphic to the direct sum of several copies of the cyclic group Z2. This may be restated in termsof the first homology group of W.The Schur multiplier M(W) (related to the second homology) was computed in (Ihara & Yokonuma 1965) for finitereflection groups and in (Yokonuma 1965) for affine reflection groups, with a more unified account given in (Howlett1988). In all cases, the Schur multiplier is also an elementary abelian 2-group. For each infinite family {Wn} offinite or affine Weyl groups, the rank of M(W) stabilizes as n goes to infinity.

4.9 See also

Artin group

Triangle group

Coxeter element

Coxeter number

Complex reflection group

ChevalleyShephardTodd theorem

Hecke algebra, a quantum deformation of the group algebra

KazhdanLusztig polynomial

Longest element of a Coxeter group

Supersoluble arrangement

https://en.wikipedia.org/wiki/Coxeter-Dynkin_diagram#Hyperbolic_Coxeter_groupshttps://en.wikipedia.org/wiki/Hyperbolic_geometryhttps://en.wikipedia.org/wiki/Triangle_grouphttps://en.wikipedia.org/wiki/Length_functionhttps://en.wikipedia.org/wiki/Word_metrichttps://en.wikipedia.org/wiki/Cayley_graphhttps://en.wikipedia.org/wiki/Cayley_graphhttps://en.wikipedia.org/wiki/Sign_maphttps://en.wikipedia.org/wiki/Partial_orderhttps://en.wikipedia.org/wiki/Weak_Bruhat_orderhttps://en.wikipedia.org/wiki/Bruhat_orderhttps://en.wikipedia.org/wiki/Fran%C3%A7ois_Bruhathttps://en.wikipedia.org/wiki/Graded_posethttps://en.wikipedia.org/wiki/Hasse_diagramhttps://en.wikipedia.org/wiki/Cayley_graphhttps://en.wikipedia.org/wiki/Abelianizationhttps://en.wikipedia.org/wiki/Elementary_abelian_grouphttps://en.wikipedia.org/wiki/Elementary_abelian_grouphttps://en.wikipedia.org/wiki/Cyclic_grouphttps://en.wikipedia.org/wiki/Group_homologyhttps://en.wikipedia.org/wiki/Schur_multiplierhttps://en.wikipedia.org/wiki/Coxeter_group#CITEREFIharaYokonuma1965https://en.wikipedia.org/wiki/Coxeter_group#CITEREFYokonuma1965https://en.wikipedia.org/wiki/Coxeter_group#CITEREFHowlett1988https://en.wikipedia.org/wiki/Coxeter_group#CITEREFHowlett1988https://en.wikipedia.org/wiki/Artin_grouphttps://en.wikipedia.org/wiki/Triangle_grouphttps://en.wikipedia.org/wiki/Coxeter_elementhttps://en.wikipedia.org/wiki/Coxeter_numberhttps://en.wikipedia.org/wiki/Complex_reflection_grouphttps://en.wikipedia.org/wiki/Chevalley%E2%80%93Shephard%E2%80%93Todd_theoremhttps://en.wikipedia.org/wiki/Hecke_algebrahttps://en.wikipedia.org/wiki/Group_algebrahttps://en.wikipedia.org/wiki/Kazhdan%E2%80%93Lusztig_polynomialhttps://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_grouphttps://en.wikipedia.org/wiki/Supersoluble_arrangement

4.10. REFERENCES 15

4.10 References[1] Brink, Brigitte; Howlett, RobertB. (1993), A finiteness property and an automatic structure for Coxeter groups, Mathe-

matische Annalen 296 (1): 179190, doi:10.1007/BF01445101, Zbl 0793.20036.

4.11 Further reading Coxeter, H. S. M. (1934), Discrete groups generated by reflections, Ann. Of Math. 35 (3): 588621, JSTOR

1968753

Coxeter, H. S. M. (1935), The complete enumeration of finite groups of the form r2i = (rirj)kij = 1 ", J.London Math. Soc., 1 10 (1): 2125, 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 ReflectionGroups, Graduate texts in mathematics 99, Springer,ISBN 978-0-387-96082-1

Humphreys, James E. (1992) [1990], Reflection Groups and Coxeter Groups, Cambridge Studies in AdvancedMathematics 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, ISBN978-0-387-98979-2, Zbl 0986.20038

Bjrner, Anders; Brenti, Francesco (2005), Combinatorics of Coxeter Groups, Graduate Texts in Mathematics231, 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):263276, doi:10.1112/jlms/s2-38.2.263, Zbl 0627.20019

Vinberg, E. B. (1984), Absence of crystallographic groups of reflections in Lobachevski spaces of large di-mension, Trudy Moskov. Mat. Obshch. 47

Ihara, S.; Yokonuma, Takeo (1965), On the second cohomology groups (Schur-multipliers) of finite reflectiongroups (PDF), Jour. Fac. Sci. Univ. Tokyo, Sect. 1 11: 155171, Zbl 0136.28802

Yokonuma, Takeo (1965), On the second cohomology groups (Schur-multipliers) of infinite discrete reflectiongroups, Jour. Fac. Sci. Univ. Tokyo, Sect. 1 11: 173186, Zbl 0136.28803

4.12 External links Hazewinkel, Michiel, ed. (2001), Coxeter group, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4

Weisstein, Eric W., Coxeter group, MathWorld.

Jenn software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators

http://link.springer.com/article/10.1007/BF01445101https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252FBF01445101https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0793.20036https://en.wikipedia.org/wiki/H._S._M._Coxeterhttps://en.wikipedia.org/wiki/JSTORhttps://www.jstor.org/stable/1968753https://en.wikipedia.org/wiki/H._S._M._Coxeterhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1112%252Fjlms%252Fs1-10.37.21http://www.math.osu.edu/~mdavis/davisbook.pdfhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-691-13138-2https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:1142.20020http://books.google.com/books?id=525Gh4uzjnIChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-387-96082-1https://en.wikipedia.org/wiki/James_E._Humphreyshttp://books.google.com/books?id=ODfjmOeNLMUChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-43613-7https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0725.20028http://books.google.com/books?id=KmL1uuiMyFUC&pg=PP10https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-387-98979-2https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0986.20038https://en.wikipedia.org/wiki/Anders_Bj%C3%B6rnerhttp://books.google.com/books?id=1TBPz5sd8m0Chttps://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematicshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-27596-1https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:1110.05001http://books.google.com/books?id=7jzvAAAAMAAJhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-273-08517-1https://en.wikipedia.org/wiki/Special:BookSources/978-0-273-08517-1https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0483.57002https://en.wikipedia.org/wiki/Nicolas_Bourbakihttp://books.google.com/books?id=2CSYFcgAlRMChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-42650-9https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0983.17001https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1112%252Fjlms%252Fs2-38.2.263https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0627.20019https://en.wikipedia.org/wiki/E._B._Vinberghttp://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6049/1/jfs110203.pdfhttp://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6049/1/jfs110203.pdfhttps://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0136.28802http://hdl.handle.net/2261/6049http://hdl.handle.net/2261/6049https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0136.28803http://www.encyclopediaofmath.org/index.php?title=p/c026980https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://mathworld.wolfram.com/CoxeterGroup.htmlhttps://en.wikipedia.org/wiki/MathWorldhttp://www.jenn3d.org/index.html

Chapter 5

Coxeter matroid

In mathematics, Coxeter matroids are generalization of matroids depending on a choice of a Coxeter group W anda parabolic subgroup P, where ordinary matroids correspond to the case when P is a maximal parabolic subgroup ofa symmetric group W. They were introduced by Gelfand and Serganova (1987, 1987b), who named them after H. S.M. Coxeter.Borovik, Gelfand & White (2003) give a detailed account of Coxeter matroids.

5.1 Definition

Suppose that W is a Coxeter group, generated by a set S of involutions, and P is a parabolic subgroup (the subgroupgenerated by some subset of S). A Coxeter matroid is a subset of W/P that for every w in W contains a uniqueminimal element with respect to the w-Bruhat order.

5.2 Relation to matroids

Suppose that the Coxeter group W is the symmetric group Sn and P is the parabolic subgroup SkSnk. Then W/Pcan be identified with the k-element subsets of the n-element set {1,2,...,n} and the elements w of W correspond tothe linear orderings of this set. A Coxeter matroid consists of k elements sets such that for each w there is a uniqueminimal element in the corresponding Bruhat ordering of k-element subsets. This is exactly the definition of a matroidof rank k on an n-element set in terms of bases: a matroid can be defined as some k-element subsets called bases ofan n-element set such that for each linear ordering of the set there is a unique minimal base in the Gale ordering ofk-element subsets.

5.3 References Borovik, Alexandre V.; Gelfand, I. M.; White, Neil (2003), Coxeter matroids, Progress in Mathematics 216,

Boston, MA: Birkhuser Boston, doi:10.1007/978-1-4612-2066-4, ISBN 978-0-8176-3764-4, MR 1989953

Gelfand, I. M.; Serganova, V. V. (1987), On the general definition of a matroid and a greedoid, DokladyAkademii Nauk SSSR (in Russian) 292 (1): 1520, ISSN 0002-3264, MR 871945

Gelfand, I. M.; Serganova, V. V. (1987b), Combinatorial geometries and the strata of a torus on homogeneouscompact manifolds, Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematich-eskikh Nauk 42 (2): 107134, doi:10.1070/RM1987v042n02ABEH001308, ISSN 0042-1316, MR 0898623 English translation in Russian Mathematical Surveys 42 (1987), no. 2, 133168

16

https://en.wikipedia.org/wiki/Matroidhttps://en.wikipedia.org/wiki/Coxeter_grouphttps://en.wikipedia.org/wiki/Parabolic_subgrouphttps://en.wikipedia.org/wiki/Coxeter_matroid#CITEREFGelfandSerganova1987https://en.wikipedia.org/wiki/Coxeter_matroid#CITEREFGelfandSerganova1987bhttps://en.wikipedia.org/wiki/H._S._M._Coxeterhttps://en.wikipedia.org/wiki/H._S._M._Coxeterhttps://en.wikipedia.org/wiki/Coxeter_matroid#CITEREFBorovikGelfandWhite2003https://en.wikipedia.org/wiki/Bruhat_orderhttps://en.wikipedia.org/wiki/Symmetric_grouphttps://en.wikipedia.org/wiki/Gale_orderinghttp://dx.doi.org/10.1007/978-1-4612-2066-4https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252F978-1-4612-2066-4https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-8176-3764-4https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1989953https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0002-3264https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=871945https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1070%252FRM1987v042n02ABEH001308https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0042-1316https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=0898623

Chapter 6

CoxeterDynkin diagram

CoxeterDynkin diagrams for the fundamental finite Coxeter groups

CoxeterDynkin diagrams for the fundamental affine Coxeter groups

In geometry, aCoxeterDynkin diagram (orCoxeter diagram,Coxeter graph) is a graph with numerically labelededges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).It describes a kaleidoscopic construction: each graph node represents a mirror (domain facet) and the label attachedto a branch encodes the dihedral angle order between two mirrors (on a domain ridge). An unlabeled branch implicitlyrepresents order-3.Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams.Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, brancheslabeled 4 or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfyan additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. Dynkindiagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.[1]

17

https://en.wikipedia.org/wiki/Geometryhttps://en.wikipedia.org/wiki/Graph_(mathematics)https://en.wikipedia.org/wiki/Mirrorhttps://en.wikipedia.org/wiki/Hyperplanehttps://en.wikipedia.org/wiki/Kaleidoscopehttps://en.wikipedia.org/wiki/Facet_(mathematics)https://en.wikipedia.org/wiki/Dihedral_anglehttps://en.wikipedia.org/wiki/Ridge_(geometry)https://en.wikipedia.org/wiki/Coxeter_grouphttps://en.wikipedia.org/wiki/Dynkin_diagramhttps://en.wikipedia.org/wiki/Directed_graphhttps://en.wikipedia.org/wiki/Undirected_graphhttps://en.wikipedia.org/wiki/Crystallographic_restriction_theoremhttps://en.wikipedia.org/wiki/Root_systemshttps://en.wikipedia.org/wiki/Semisimple_Lie_algebra

18 CHAPTER 6. COXETERDYNKIN DIAGRAM

6.1 Description

Branches of a CoxeterDynkin diagram are labeled with a rational number p, representing a dihedral angle of 180/p.When p = 2 the angle is 90 and the mirrors have no interaction, so the branch can be omitted from the diagram.If a branch is unlabeled, it is assumed to have p = 3, representing an angle of 60. Two parallel mirrors have abranch marked with "". In principle, n mirrors can be represented by a complete graph in which all n(n 1) / 2branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, sothe corresponding branches are omitted.Diagrams can be labeled by their graph structure. The first studied forms by Ludwig Schlfli are the orthoschemes aslinear and generate regular polytopes and regular honeycombs. Plagioschemes are simplices represented by branch-ing graphs, and cycloschemes are simplices represented by cyclic graphs.

6.2 Schlfli matrix

Every Coxeter diagram has a corresponding Schlfli matrix with matrix elements a, = a , = 2cos ( / p) where p isthe branch order between the pairs of mirrors. As a matrix of cosines, it is also called a Gramian matrix after JrgenPedersen Gram. All Coxeter group Schlfli matrices are symmetric because their root vectors are normalized. It isrelated closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the limited cases of p= 2,3,4, and 6, which are NOT symmetric in general.The determinant of the Schlfli matrix, called the Schlflian, and its sign determines whether the group is finite(positive), affine (zero), indefinite (negative). This rule is called Schlflis Criterion.[2]

The eigenvalues of the Schlfli matrix determines whether a Coxeter group is of finite type (all positive), affine type (allnon-negative, at least one is 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 hyperbolicCoxeter groups. We use the following definition: A Coxeter group with connected diagram is hyperbolic if it is neitherof finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter groupis compact if all subgroups are finite (i.e. have positive determinants), and paracompact if all its subgroups are finiteor affine (i.e. have nonnegative determinants).Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannrand F. Lannr who enumerated the compact hyperbolic groups in 1950,[3] and Koszul (or quasi-Lannr) for theparacompact groups.

6.2.1 Rank 2 Coxeter groups

For rank 2, the type of a Coxeter group is fully determined by the determinant of the Schlfli matrix, as it is simply theproduct of the eigenvalues: Finite type (positive determinant), affine type (zero determinant) or hyperbolic (negativedeterminant). Coxeter uses an equivalent bracket notation which lists sequences of branch orders as a substitute forthe node-branch graphic diagrams.

6.2.2 Geometric visualizations

The CoxeterDynkin diagram can be seen as a graphic description of the fundamental domain of mirrors. A mirrorrepresents a hyperplane within a given dimensional spherical or Euclidean or hyperbolic space. (In 2D spaces, amirror is a line, and in 3D a mirror is a plane).These visualizations show the fundamental domains for 2D and 3D Euclidean groups, and 2D spherical groups. Foreach the Coxeter diagram can be deduced by identifying the hyperplane mirrors and labelling their connectivity,ignoring 90-degree dihedral angles (order 2).

6.3 Finite Coxeter groups

See also polytope families for a table of end-node uniform polytopes associated with these groups.

https://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Dihedral_anglehttps://en.wikipedia.org/wiki/Complete_graphhttps://en.wikipedia.org/wiki/Ludwig_Schl%C3%A4flihttps://en.wikipedia.org/wiki/Orthoschemehttps://en.wikipedia.org/wiki/Regular_polytopehttps://en.wikipedia.org/wiki/Regular_honeycombhttps://en.wikipedia.org/wiki/Ludwig_Schl%C3%A4flihttps://en.wikipedia.org/wiki/Gramian_matrixhttps://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gramhttps://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gramhttps://en.wikipedia.org/wiki/Coxeter_grouphttps://en.wikipedia.org/wiki/Cartan_matrixhttps://en.wikipedia.org/wiki/Dynkin_diagramhttps://en.wikipedia.org/wiki/Eigenvalueshttps://en.wikipedia.org/wiki/Coxeter_notationhttps://en.wikipedia.org/wiki/Fundamental_domainhttps://en.wikipedia.org/wiki/Hyperplanehttps://en.wikipedia.org/wiki/Polytope_families

6.4. APPLICATION WITH UNIFORM POLYTOPES 19

Three different symbols are given for the same groups as a letter/number, as a bracketed set of numbers, andas the Coxeter diagram.

The bifurcated Dn groups is half or alternated version of the regular Cn groups.

The bifurcated Dn and En groups are also labeled by a superscript form [3a,b,c] where a,b,c are the numbers ofsegments in each of the three branches.

6.4 Application with uniform polytopes

CoxeterDynkin diagrams can explicitly enumerate nearly all classes of uniform polytope and uniform tessellations.Every uniform polytope with pure reflective symmetry (all but a few special cases have pure reflectional symmetry) canbe represented by a CoxeterDynkin diagram with permutations ofmarkups. Each uniform polytope can be generatedusing such mirrors and a single generator point: mirror images create new points as reflections, then polytope edgescan be defined between points and a mirror image point. Faces can be constructed by cycles of edges created, etc.To specify the generating vertex, one or more nodes are marked with rings, meaning that the vertex is not on themirror(s) represented by the ringed node(s). (If two or more mirrors are marked, the vertex is equidistant fromthem.) A mirror is active (creates reflections) only with respect to points not on it. A diagram needs at least one activenode to represent a polytope.All regular polytopes, represented by Schlfli symbol symbol {p, q, r, ...}, can have their fundamental domainsrepresented by a set of n mirrors with a related CoxeterDynkin diagram of a line of nodes and branches labeled byp, q, r, ..., with the first node ringed.Uniform polytopes with one ring correspond to generator points at the corners of the fundamental domain simplex.Two rings correspond to the edges of simplex and have a degree of freedom, with only the midpoint as the uniformsolution for equal edge lengths. In general k-rings generators are on k-faces of the simplex, and if all the nodes areringed, the generator point is in the interior of the simplex.A secondary markup conveys a special case nonreflectional symmetry uniform polytopes. These cases exist asalternations of reflective symmetry polytopes. This markup removes the central dot of a ringed node, called a hole(circles with nodes removed), to imply alternate nodes deleted. The resulting polytope will have a subsymmetry ofthe original Coxeter group. A truncated alternation is called a snub.

A single node represents a single mirror. This is called group A1. If ringed this creates a line segment perpen-dicular to the mirror, represented as {}.

Two unattached nodes represent two perpendicular mirrors. If both nodes are ringed, a rectangle can becreated, or a square if the point is at equal distance from both mirrors.

Two nodes attached by an order-n branch can create an n-gon if the point is on one mirror, and a 2n-gon if thepoint is off both mirrors. This forms the I1(n) group.

Two parallel mirrors can represent an infinite polygon I1() group, also called 1.

Three mirrors in a triangle form images seen in a traditional kaleidoscope and can be represented by threenodes connected in a triangle. Repeating examples will have branches labeled as (3 3 3), (2 4 4), (2 3 6),although the last two can be drawn as a line (with the 2 branches ignored). These will generate uniform tilings.

Three mirrors can generate uniform polyhedra; including rational numbers gives the set of Schwarz triangles.

Three mirrors with one perpendicular to the other two can form the uniform prisms.

The duals of the uniform polytopes are sometimes marked up with a perpendicular slash replacing ringed nodes, and

a slash-hole for hole nodes of the snubs. For example represents a rectangle (as two active orthogonal mirrors),

and represents its dual polygon, the rhombus.

https://en.wikipedia.org/wiki/Uniform_polytopehttps://en.wikipedia.org/wiki/Uniform_tilinghttps://en.wikipedia.org/wiki/Edge_(geometry)https://en.wikipedia.org/wiki/Face_(geometry)https://en.wikipedia.org/wiki/Regular_polytopehttps://en.wikipedia.org/wiki/Schl%C3%A4fli_symbolhttps://en.wikipedia.org/wiki/Fundamental_domainhttps://en.wikipedia.org/wiki/Alternation_(geometry)https://en.wikipedia.org/wiki/Coxeter_grouphttps://en.wikipedia.org/wiki/Snub_(geometry)https://en.wikipedia.org/wiki/Line_segmenthttps://en.wikipedia.org/wiki/Perpendicularhttps://en.wikipedia.org/wiki/Rectanglehttps://en.wikipedia.org/wiki/Square_(geometry)https://en.wikipedia.org/wiki/Polygonhttps://en.wikipedia.org/wiki/Kaleidoscopehttps://en.wikipedia.org/wiki/Tiling_by_regular_polygonshttps://en.wikipedia.org/wiki/Uniform_polyhedron#(4_3_2)_Oh_Octahedral_symmetryhttps://en.wikipedia.org/wiki/Schwarz_trianglehttps://en.wikipedia.org/wiki/Prism_(geometry)https://en.wikipedia.org/wiki/Rectanglehttps://en.wikipedia.org/wiki/Dual_polygonhttps://en.wikipedia.org/wiki/Rhombus

20 CHAPTER 6. COXETERDYNKIN DIAGRAM

In constructing uniform polytopes, nodes are marked as active by a ring if a generator point is off the mirror, creating a new edgebetween a generator point and its mirror image. An unringed node represents an inactive mirror that generates no new points.

6.5. AFFINE COXETER GROUPS 21

6.4.1 Example polyhedra and tilings

For example, the B3 Coxeter group has a diagram: . This is also called octahedral symmetry.There are 7 convex uniform polyhedra that can be constructed from this symmetry group and 3 from its alternationsubsymmetries, each with a uniquely marked up CoxeterDynkin diagram. The Wythoff symbol represents a specialcase of the Coxeter diagram for rank 3 graphs, with all 3 branch orders named, rather than suppressing the order 2branches. The Wythoff symbol is able to handle the snub form, but not general alternations without all nodes ringed.The same constructions can be made on disjointed (orthogonal) Coxeter groups like the uniform prisms, and can beseen more clearly as tilings of dihedrons and hosohedrons on the sphere, like this [6][] or [6,2] family:

In comparison the [6,3], family produces a parallel set of 7 uniform tilings of the Euclidean plane, and theirdual tilings. There are again 3 alternations and some half symmetry version.

In the hyperbolic plane [7,3], family produces a parallel set of uniform tilings of the Euclidean plane, and theirdual tilings. There is only 1 alternation (snub) since all branch orders are odd. Many other hyperbolic families ofuniform tilings can be seen at uniform tilings in hyperbolic plane.

6.5 Affine Coxeter groups

Families of convex uniform Euclidean tessellations are defined by the affine Coxeter groups. These groups are iden-tical to the finite groups with the inclusion of one added node. In letter names they are given the same letter with a"~" above the letter. The index refers to the finite group, so the rank is the index plus 1. (Ernst Witt symbols for theaffine groups are given as also)

1. An1 : diagrams of this type are cycles. (Also P )

2. Cn1 is associated with the hypercube regular tessellation {4, 3, ...., 4} family. (Also R )

3. Bn1 related to C by one removed mirror. (Also S )

4. Dn1 related to C by two removed mirrors. (Also Q )

5. E6 , E7 , E8 . (Also T7, T8, T9)

6. F4 forms the {3,4,3,3} regular tessellation. (Also U5)

7. G2 forms 30-60-90 triangle fundamental domains. (Also V3)

8. I1 is two parallel mirrors. ( = A1 = C1 ) (Also W2)

Composite groups can also be defined as orthogonal projects. The most common use A1 , like A21 , represents

square or rectangular checker board domains in the Euclidean plane. And A1G2 represents triangular prismfundamental domains in Euclidean 3-space.

6.6 Hyperbolic Coxeter groups

There are many infinite hyperbolic Coxeter groups. Hyperbolic groups are categorized as compact or not, withcompact groups having bounded fundamental domains. Compact simplex hyperbolic groups (Lannr simplices)exist as rank 3 to 5. Paracompact simplex groups (Koszul simplices) exist up to rank 10. Hypercompact (Vinbergpolytopes) groups have been explored but not been fully determined. In 2006, Allcock proved that there are infinitelymany compact Vinberg polytopes for dimension up to 6, and infinitely many finite-volume Viberg polytopes fordimension up to 19,[4] so a complete enumeration is not possible. All of these fundamental reflective domains, bothsimplices and nonsimplices, are often called Coxeter polytopes or sometimes less accurately Coxeter polyhedra.

https://en.wikipedia.org/wiki/Coxeter_grouphttps://en.wikipedia.org/wiki/Octahedral_symmetryhttps://en.wikipedia.org/wiki/Uniform_polyhedrahttps://en.wikipedia.org/wiki/Alternation_(geometry)https://en.wikipedia.org/wiki/Wythoff_symbolhttps://en.wikipedia.org/wiki/Prism_(geometry)https://en.wikipedia.org/wiki/Dihedronhttps://en.wikipedia.org/wiki/Hosohedronhttps://en.wikipedia.org/wiki/Snub_(geometry)https://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_planehttps://en.wikipedia.org/wiki/Affine_Coxeter_grouphttps://en.wikipedia.org/wiki/Ernst_Witthttps://en.wikipedia.org/wiki/Hypercubic_honeycombhttps://en.wikipedia.org/wiki/Checker_boardhttps://en.wikipedia.org/wiki/Triangular_prismhttps://en.wikipedia.org/wiki/Coxeter_grouphttps://en.wikipedia.org/wiki/Polytopehttps://en.wikipedia.org/wiki/Polyhedron

22 CHAPTER 6. COXETERDYNKIN DIAGRAM

6.6.1 Hyperbolic groups in H2

Further information: Uniform tilings in hyperbolic plane

Two-dimensional hyperbolic triangle groups exists as rank 3 Coxeter diagrams, defined by triangle (p q r) for:

1

p+

1

q+

1

r< 1.

There are infinitely many compact triangular hyperbolic Coxeter groups, including linear and triangle graphs. Thelinear graphs exist for right triangles (with r=2).[5]

Paracompact Coxeter groups of rank 3 exist as limits to the compact ones.

Arithmetic triangle group

A finite subset of hyperbolic triangle groups are arithmetic groups. By computer search the complete list was de-termined by Kisao Takeuchi in his 1977 paper Arithmetic triangle groups.[6] There are 85 total, 76 compact and 9paracompact.

Hyperbolic Coxeter polygons above triangles

Other H2 hyperbolic kaleidoscopes can be constructed from higher order polygons. Like triangle groups these kalei-doscopes can be identified by a cyclic sequence of mirror intersection orders around the fundamental domain, as (a bc d ...), or equivalently in orbifold notation as *abcd.... CoxeterDynkin diagrams for these polygonal kaleidoscopescan be seen as a degenerate (n-1)-simplex fundamental domains, with a cyclic of branches order a,b,c... and theremaining n*(n-3)/2 branches are labeled as infinite () representing the non-intersecting mirrors. The only nonhy-

perbolic example is Euclidean symmetry four mirrors in a square or rectangle as , [,2,] (orbifold *2222).Another branch representation for non-intersecting mirrors by Vinberg gives infinite branches as dotted or dashed

lines, so this diagram can be shown as , with the four order-2 branches suppressed around the perimeter.For example a quadrilateral domain (a b c d) will have two infinite order branches connecting ultraparallel mirrors.

The smallest hyperbolic example is , [,3,] or [i/1,3,i/2] (orbifold *3222), where (1,2) are the

distance between the ultraparallel mirrors. The alternate expression is , with three order-2 branches suppressed

around the perimeter. Similarly (2 3 2 3) (orbifold *3232) can be represented as and (3 3 3 3), (orbifold *3333)

can be represented as a complete graph .The highest quadrilateral domain ( ) is an infinite square, represented by a complete tetrahedral graph with4 perimeter branches as ideal vertices and two diagonal branches as infinity (shown as dotted lines) for ultraparallel

mirrors: .

6.6.2 Compact (Lannr simplex groups)

Compact hyperbolic groups are called Lannr groups after Folke Lannr who first studied them in 1950.[7] They onlyexist as rank 4 and 5 graphs. Coxeter studied the linear hyperbolic coxeter groups in his 1954 paper Regular Honey-

combs in hyperbolic space,[8] which included two rational solutions in hyperbolic 4-space: [5/2,5,3,3] =

and [5,5/2,5,3] = .

https://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_planehttps://en.wikipedia.org/wiki/Triangle_grouphttps://en.wikipedia.org/wiki/Triangle_grouphttps://en.wikipedia.org/wiki/Arithmetic_grouphttps://en.wikipedia.org/wiki/Triangle_grouphttps://en.wikipedia.org/wiki/Orbifold_notationhttps://en.wikipedia.org/wiki/Simplexhttps://en.wikipedia.org/wiki/Square_tilinghttps://en.wikipedia.org/wiki/Ernest_Borisovich_Vinberghttps://en.wikipedia.org/wiki/Tetrahedronhttps://en.wikipedia.org/wiki/Ultraparallel_theoremhttps://en.wikipedia.org/wiki/Folke_Lann%C3%A9rhttps://en.wikipedia.org/wiki/Order-5_5-cell_honeycomb

6.6. HYPERBOLIC COXETER GROUPS 23

Ranks 45

Further information: Uniform honeycombs in hyperbolic space

The fundamental domain of either of the two bifurcating groups, [5,31,1] and [5,3,31,1], is double that of a cor-responding linear group, [5,3,4] and [5,3,3,4] respectively. Letter names are given by Johnson as extended Wittsymbols.[9]

6.6.3 Paracompact (Koszul simplex groups)

An example order-3 apeirogonal tiling, {,3} with one green apeirogon and its circumscribed horocycle

Paracompact (also called noncompact) hyperbolic Coxeter groups contain affine subgroups and have asymptotic sim-plex fundamental domains. The highest paracompact hyperbolic Coxeter group is rank 10. These groups are namedafter French mathematician Jean-Louis Koszul.[10] They are also called quasi-Lannr groups extending the compactLannr groups. The list was determined complete by computer search by M. Chein and published in 1969.[11]

By Vinberg, all but eight of these 72 compact and paracompact simplices are arithmetic. Two of the nonarith-

metic groups are compact: and . The other six nonarithmetic groups are all paracompact, with five

https://en.wikipedia.org/wiki/Uniform_honeycombs_in_hyperbolic_spacehttps://en.wikipedia.org/wiki/Norman_Johnson_(mathematician)https://en.wikipedia.org/wiki/Ernst_Witthttps://en.wikipedia.org/wiki/Ernst_Witthttps://en.wikipedia.org/wiki/Order-3_apeirogonal_tilinghttps://en.wikipedia.org/wiki/Apeirogonhttps://en.wikipedia.org/wiki/Horocyclehttps://en.wikipedia.org/wiki/Jean-Louis_Koszul

24 CHAPTER 6. COXETERDYNKIN DIAGRAM

3-dimensional groups , , , , and , and one 5-dimensional group .

Ideal simplices

Ideal fundamental domains of , [(,,)] seen in the Poincare disk model

There are 5 hyperbolic Coxeter groups expressing ideal simplices, graphs where removal of any one node results inan affine Coxeter group. Thus all vertices of this ideal simplex are at infinity.[12]

Ranks 410

Further information: Paracompact uniform honeycombs

There are a total of 58 paracompact hyperbolic Coxeter groups from rank 4 through 10. All 58 are grouped belowin five categories. Letter symbols are given by Johnson as Extended Witt symbols, using PQRSTWUV from the affineWitt symbols, and adding LMNOXYZ. These hyperbolic groups are given an overline, or a hat, for cycloschemes.The bracket notation from Coxeter is a linearized representation of the Coxeter group.

https://en.wikipedia.org/wiki/Poincare_disk_modelhttps://en.wikipedia.org/wiki/Paracompact_uniform_honeycombshttps://en.wikipedia.org/wiki/Norman_Johnson_(mathematician)https://en.wikipedia.org/wiki/Coxeter_notation

6.6. HYPERBOLIC COXETER GROUPS 25

Infinite Euclidean cells like a hexagonal tiling, properly scaled converge to a single ideal point at infinity, like the hexagonal tilinghoneycomb, {6,3,3}, as shown with this single cell in a Poincar disk model projection.

Subgroup relations of paracompact hyperbolic groups These trees represents subgroup relations of paracom-pact hyperbolic groups. Subgroup indices on each connection are given in red.[13] Subgroups of index 2 representa mirror removal, and fundamental domain doubling. Others can be inferred by commensurability (integer ratio ofvolumes) for the tetrahedral domains.

6.6.4 Hypercompact Coxeter groups (Vinberg polytopes)

Just like the hyperbolic plane H2 has nontrianglar polygonal domains, higher-dimensional reflective hyperbolic do-mains also exists. These nonsimplex domains can be considered degenerate simplices with non-intersecting mirrorsgiven infinite order, or in a Coxeter diagram, such branches are given dotted or dashed lines. These nonsimplexdomains are called Vinberg polytopes, after Ernest Vinberg for his Vinbergs algorithm for finding nonsimplex fun-damental domain of a hyperbolic reflection group. Geometrically these fundamental domains can be classified asquadrilateral pyramids, or prisms or other polytopes with all edges having dihedral angles as /n for n=2,3,4...In a simplex-based domain, there are n+1 mirrors for n-dimensional space. In non-simplex domains, there are morethan n+1 mirrors. The list is finite, but not completely known. Instead partial lists have been enumerated as n+kmirrors for k as 2,3, and 4.

https://en.wikipedia.org/wiki/Hexagonal_tilinghttps://en.wikipedia.org/wiki/Hexagonal_tiling_honeycombhttps://en.wikipedia.org/wiki/Hexagonal_tiling_honeycombhttps://en.wikipedia.org/wiki/Poincar%C3%A9_disk_modelhttps://en.wikipedia.org/wiki/Commensurability_(mathematics)https://en.wikipedia.org/wiki/Ernest_Vinberghttps://en.wikipedia.org/wiki/Vinberg%2527s_algorithmhttps://en.wikipedia.org/wiki/Pyramid_(geometry)https://en.wikipedia.org/wiki/Prism_(geometry)https://en.wikipedia.org/wiki/Polytopehttps://en.wikipedia.org/wiki/Dihedral_angle

26 CHAPTER 6. COXETERDYNKIN DIAGRAM

Hypercompact Coxeter groups in three dimensional space or higher differ from two dimensional groups in one essen-tial respect. Two hyperbolic n-gons having the same angles in the same cyclic order may have different edge lengthsand are not in general congruent. In contrast Vinberg polytopes in 3 dimensions or higher are completely determinedby the dihedral angles. This fact is based on the Mostow rigidity theorem, that two isomorphic groups generated byreflections in Hn for n>=3, define congruent fundamental domains (Vinberg polytopes).

Vinberg polytopes with rank n+2 for n dimensional space

The complete list of compact hyperbolic Vinberg polytopes with rank n+2 mirrors for n-dimensions has been enu-merated by F. Esselmann in 1996.[14] A partial list was published in 1974 by I. M. Kaplinskaya.[15]

The complete list of paracompact solutions was published by P. Tumarkin in 2003, with dimensions from 3 to 17.[16]

The smallest paracompact form in H3 can be represented by , or [,3,3,] which can be constructedby a mirror removal of paracompact hyperbolic g