Coxeter–Dynkin diagram


In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents a mirror and the label attached to a branch encodes the dihedral angle order between two mirrors, that is, the amount by which the angle between the reflective planes can be multiplied by to get 180 degrees. An unlabeled branch implicitly represents 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, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.

Description

Branches of a Coxeter–Dynkin diagram are labeled with a rational number p, representing a dihedral angle of 180°/p. When 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, representing an angle of 60°. Two parallel mirrors have a branch marked with "∞". In principle, n mirrors can be represented by a complete graph in which all n( branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, so the corresponding branches are omitted.
Diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes and regular honeycombs. Plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs.

Schläfli matrix

Every Coxeter diagram has a corresponding Schläfli matrix
, with matrix elements where p is the branch order between the pairs of mirrors. As a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related 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 Schläfli matrix, called the Schläflian, and its sign determines whether the group is finite, [|affine], indefinite. This rule is called Schläfli's Criterion.
The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of finite type, affine type, or indefinite type. The indefinite type is sometimes further subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups. We use the following definition: A Coxeter group with connected diagram is hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact if all subgroups are finite, and paracompact if all its subgroups are finite or affine.
Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannér, after F. Lannér who enumerated the compact hyperbolic groups in 1950, and Koszul for the paracompact groups.

Rank 2 Coxeter groups

For rank 2, the type of a Coxeter group is fully determined by the determinant of the Schläfli matrix, as it is simply the product of the eigenvalues: Finite type, affine type or hyperbolic. Coxeter uses an equivalent bracket notation which lists sequences of branch orders as a substitute for the node-branch graphic diagrams. Rational solutions ,, also exist, with gcd=1, which define overlapping fundamental domains. For example, 3/2, 4/3, 5/2, 5/3, 5/4. and 6/5.

Geometric visualizations

The Coxeter–Dynkin diagram can be seen as a graphic description of the fundamental domain of mirrors. A mirror represents a hyperplane within a given dimensional spherical or Euclidean or hyperbolic space..
These visualizations show the fundamental domains for 2D and 3D Euclidean groups, and 2D spherical groups. For each the Coxeter diagram can be deduced by identifying the hyperplane mirrors and labelling their connectivity, ignoring 90-degree dihedral angles.

Finite Coxeter groups

Coxeter–Dynkin diagrams can explicitly enumerate nearly all classes of uniform polytope and uniform tessellations. Every uniform polytope with pure reflective symmetry can be represented by a Coxeter–Dynkin diagram with permutations of markups. Each uniform polytope can be generated using such mirrors and a single generator point: mirror images create new points as reflections, then polytope edges can be defined between points and a mirror image point. Faces are generated by the repeated reflection of an edge eventually wrapping around to the original generator; the final shape, as well as any higher-dimensional facets, are likewise created by the face being reflected to enclose an area.
To specify the generating vertex, one or more nodes are marked with rings, meaning that the vertex is not on the mirror represented by the ringed node. A mirror is active only with respect to points not on it. A diagram needs at least one active node to represent a polytope. An unconnected diagram requires at least one active node in each subgraph.
All regular polytopes, represented by Schläfli symbol, can have their fundamental domains represented by a set of n mirrors with a related Coxeter–Dynkin diagram of a line of nodes and branches labeled by 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 uniform solution for equal edge lengths. In general k-ring generator points are on -faces of the simplex, and if all the nodes are ringed, the generator point is in the interior of the simplex.
The special case of uniform polytopes with non-reflectional symmetry is represented by a secondary markup where the central dot of a ringed node is removed. These shapes are alternations of polytopes with reflective symmetry, implying that alternate nodes are deleted. The resulting polytope will have a subsymmetry of the original Coxeter group. A truncated alternation is called a snub.

There are 7 reflective uniform constructions within a general triangle, based on 7 topological generator positions within the fundamental domain. Every active mirror generates an edge, with two active mirrors have generators on the domain sides and three active mirrors has the generator in the interior. One or two degrees of freedom can be solved for a unique position for equal edge lengths of the resulting polyhedron or tiling.
Example 7 generators on octahedral symmetry, fundamental domain triangle, with 8th snub generation as an alternation

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, and represents its dual polygon, the rhombus.

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 alternation subsymmetries, each with a uniquely marked up Coxeter–Dynkin diagram. The Wythoff symbol represents a special case of the Coxeter diagram for rank 3 graphs, with all 3 branch orders named, rather than suppressing the order 2 branches. 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 Coxeter groups like the uniform prisms, and can be seen more clearly as tilings of dihedrons and hosohedrons on the sphere, like this × or family:
In comparison, the , family produces a parallel set of 7 uniform tilings of the Euclidean plane, and their dual tilings. There are again 3 alternations and some half symmetry version.
In the hyperbolic plane , family produces a parallel set of uniform tilings, and their dual tilings. There is only 1 alternation since all branch orders are odd. Many other hyperbolic families of uniform tilings can be seen at uniform tilings in hyperbolic plane.

Affine Coxeter groups

Families of convex uniform Euclidean tessellations are defined by the affine Coxeter groups. These groups are identical 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.
  1. : diagrams of this type are cycles.
  2. is associated with the hypercube regular tessellation family.
  3. related to C by one removed mirror.
  4. related to C by two removed mirrors.
  5. ,,.
  6. forms the regular tessellation.
  7. forms 30-60-90 triangle fundamental domains.
  8. is two parallel mirrors.
Composite groups can also be defined as orthogonal projects. The most common use, like, represents square or rectangular checker board domains in the Euclidean plane. And represents triangular prism fundamental domains in Euclidean 3-space.
Rank / /
2=
 =
3=]
:File:DynkinA2Affine.svg|* 		=
:File:DynkinC2Affine.svg|* 		=
:File:DynkinG2Affine1.svg|*
4=]
:File:DynkinA3Affine.svg|* 		=
:File:DynkinB3Affine.svg|* 		=
:File:DynkinC3Affine.svg|* 		=
=
5=]
:File:DynkinA4Affine.svg|* 		=
:File:DynkinB4Affine.svg|* 		=
:File:DynkinC4Affine.svg|* 		=
:File:DynkinD4Affine.svg|* 		=
:File:DynkinF4Affine.svg|*
6=]
:File:DynkinA5Affine1.svg|* 		=
:File:DynkinB5Affine.svg|* 		=
:File:DynkinC5Affine.svg|* 		=
:File:DynkinD5Affine.svg|*
7=]
:File:AffineA6.svg|* 		=
=
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8=]
:File:AffineA7.svg|* 		=
:File:AffineB7.svg|* 		=
=
:File:AffineD7.svg|* 		=
:File:AffineE7.svg|*
9=]
:File:AffineA8.svg|* 		=
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:File:E9-AffineE8.svg|*
10=]
:File:AffineA9.svg|* 		=
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-
11............-

Hyperbolic Coxeter groups

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

Hyperbolic groups in H2

Two-dimensional hyperbolic triangle groups exist as rank 3 Coxeter diagrams, defined by triangle for:
There are infinitely many compact triangular hyperbolic Coxeter groups, including linear and triangle graphs. The linear graphs exist for right triangles.
LinearCyclic--
∞ , :
2<pq
...
...
...
∞ , : p+q+r>9
Paracompact Coxeter groups of rank 3 exist as limits to the compact ones.
Linear graphsCyclic graphs

    • Arithmetic triangle group

    The hyperbolic triangle groups that are also arithmetic groups form a finite subset. By computer search the complete list was determined by Kisao Takeuchi in his 1977 paper Arithmetic triangle groups. There are 85 total, 76 compact and 9 paracompact.
    Right triangles General triangles
    Compact groups:
    Paracompact right triangles:
    Paracompact general triangles:

    Hyperbolic Coxeter polygons above triangles

    Other H2 hyperbolic kaleidoscopes can be constructed from higher order polygons. Like triangle groups these kaleidoscopes can be identified by a cyclic sequence of mirror intersection orders around the fundamental domain, as, or equivalently in orbifold notation as *abcd.... Coxeter–Dynkin diagrams for these polygonal kaleidoscopes can be seen as a degenerate -simplex fundamental domains, with a cyclic of branches order a,b,c... and the remaining n*/2 branches are labeled as infinite representing the non-intersecting mirrors. The only nonhyperbolic example is Euclidean symmetry four mirrors in a square or rectangle as, . 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 will have two infinite order branches connecting ultraparallel mirrors. The smallest hyperbolic example is, or , where are the distance between the ultraparallel mirrors. The alternate expression is, with three order-2 branches suppressed around the perimeter. Similarly can be represented as and, can be represented as a complete graph.
    The highest quadrilateral domain is an infinite square, represented by a complete tetrahedral graph with 4 perimeter branches as ideal vertices and two diagonal branches as infinity for ultraparallel mirrors:.

    Compact (Lannér simplex groups)

    Compact hyperbolic groups are called Lannér groups after Folke Lannér who first studied them in 1950. They only exist as rank 4 and 5 graphs. Coxeter studied the linear hyperbolic coxeter groups in his 1954 paper Regular Honeycombs in hyperbolic space, which included two rational solutions in hyperbolic 4-space: = and =.

    Ranks 4–5

    The fundamental domain of either of the two bifurcating groups, and , is double that of a corresponding linear group, and respectively. Letter names are given by Johnson as extended Witt symbols.
    Dimension
    Hd    		RankTotal countLinearBifurcatingCyclic
    H349
    3:

    = :
    = :
    = :
    = :
    = :
    = :
    = ]:
    = :
    = ]:
    H455
    3:

    = :
    = :
    = :
    = :
    = :

    Paracompact (Koszul simplex groups)

    Paracompact hyperbolic Coxeter groups contain affine subgroups and have asymptotic simplex fundamental domains. The highest paracompact hyperbolic Coxeter group is rank 10. These groups are named after French mathematician Jean-Louis Koszul. They are also called quasi-Lannér groups extending the compact Lannér groups. The list was determined complete by computer search by M. Chein and published in 1969.
    By Vinberg, all but eight of these 72 compact and paracompact simplices are arithmetic. Two of the nonarithmetic groups are compact: and. The other six nonarithmetic groups are all paracompact, with five 3-dimensional groups,,,, and, and one 5-dimensional group.

    Ideal simplices

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

    Ranks 4–10

    There are a total of 58 paracompact hyperbolic Coxeter groups from rank 4 through 10. All 58 are grouped below in five categories. Letter symbols are given by Johnson as Extended Witt symbols, using PQRSTWUV from the affine Witt 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.
    Subgroup relations of paracompact hyperbolic groups
    These trees represents subgroup relations of paracompact hyperbolic groups. Subgroup indices on each connection are given in red. Subgroups of index 2 represent a mirror removal, and fundamental domain doubling. Others can be inferred by commensurability for the tetrahedral domains.

    Hypercompact Coxeter groups (Vinberg polytopes)

    Just like the hyperbolic plane H2 has nontriangular polygonal domains, higher-dimensional reflective hyperbolic domains also exists. These nonsimplex domains can be considered degenerate simplices with non-intersecting mirrors given infinite order, or in a Coxeter diagram, such branches are given dotted or dashed lines. These nonsimplex domains are called Vinberg polytopes, after Ernest Vinberg for his Vinberg's algorithm for finding nonsimplex fundamental domain of a hyperbolic reflection group. Geometrically these fundamental domains can be classified as quadrilateral pyramids, or prisms or other polytopes with edges as the intersection of two mirrors having dihedral angles as π/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 more than n+1 mirrors. The list is finite, but not completely known. Instead partial lists have been enumerated as n+k mirrors for k as 2,3, and 4.
    Hypercompact Coxeter groups in three dimensional space or higher differ from two dimensional groups in one essential respect. Two hyperbolic n-gons having the same angles in the same cyclic order may have different edge lengths and are not in general congruent. In contrast Vinberg polytopes in 3 dimensions or higher are completely determined by the dihedral angles. This fact is based on the Mostow rigidity theorem, that two isomorphic groups generated by reflections in Hn for n>=3, define congruent fundamental domains.

    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 enumerated by F. Esselmann in 1996. A partial list was published in 1974 by I. M. Kaplinskaya.
    The complete list of paracompact solutions was published by P. Tumarkin in 2003, with dimensions from 3 to 17.
    The smallest paracompact form in H3 can be represented by, or which can be constructed by a mirror removal of paracompact hyperbolic group as . The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramids include = , =. Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow-tie graphs: = or, = or, = or.
    Other valid paracompact graphs with quadrilateral pyramid fundamental domains include:
    DimensionRankGraphs
    H35
    Another subgroup = = ]. = =.

    Vinberg polytopes with rank n+3 for n dimensional space

    There are a finite number of degenerate fundamental simplices exist up to 8-dimensions. The complete list of Compact Vinberg polytopes with rank n+3 mirrors for n-dimensions has been enumerated by P. Tumarkin in 2004. These groups are labeled by dotted/broken lines for ultraparallel branches. The complete list of non-Compact Vinberg polytopes with rank n+3 mirrors and with one non-simple vertex for n-dimensions has been enumerated by Mike Roberts.
    For 4 to 8 dimensions, rank 7 to 11 Coxeter groups are counted as 44, 16, 3, 1, and 1 respectively. The highest was discovered by Bugaenko in 1984 in dimension 8, rank 11:

    Vinberg polytopes with rank n+4 for n dimensional space

    There are a finite number of degenerate fundamental simplices exist up to 8-dimensions. Compact Vinberg polytopes with rank n+4 mirrors for n-dimensions has been explored by A. Felikson and P. Tumarkin in 2005.

    Lorentzian groups

    Lorentzian groups for simplex domains can be defined as graphs beyond the paracompact hyperbolic forms. These are sometimes called super-ideal simplices and are also related to a Lorentzian geometry, named after Hendrik Lorentz in the field of special and general relativity space-time, containing one time-like dimensional components whose self dot products are negative. Danny Calegari calls these convex cocompact Coxeter groups in n-dimensional hyperbolic space.
    A 1982 paper by George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, enumerates the finite list of Lorentzian of rank 5 to 11. He calls them level 2, meaning removal any permutation of 2 nodes leaves a finite or Euclidean graph. His enumeration is complete, but didn't list graphs that are a subgroup of another. All higher-order branch Coxeter groups of rank-4 are Lorentzian, ending in the limit as a complete graph 3-simplex Coxeter-Dynkin diagram with 6 infinite order branches, which can be expressed as ]. Rank 5-11 have a finite number of groups 186, 66, 36, 13, 10, 8, and 4 Lorentzian groups respectively. A 2013 paper by H. Chen and J.-P. Labbé, Lorentzian Coxeter groups and Boyd--Maxwell ball packings, recomputed and published the complete list.
    For the highest ranks 8-11, the complete lists are:

    Very-extended Coxeter Diagrams

    One usage includes a very-extended definition from the direct Dynkin diagram usage which considers affine groups as extended, hyperbolic groups over-extended, and a third node as very-extended simple groups. These extensions are usually marked by an exponent of 1,2, or 3 + symbols for the number of extended nodes. This extending series can be extended backwards, by sequentially removing the nodes from the same position in the graph, although the process stops after removing branching node. The E8 extended family is the most commonly shown example extending backwards from E3 and forwards to E11.
    The extending process can define a limited series of Coxeter graphs that progress from finite to affine to hyperbolic to Lorentzian. The determinant of the Cartan matrices determine where the series changes from finite to affine to hyperbolic, and ending as a Lorentzian group, containing at least one hyperbolic subgroup. The noncrystalographic Hn groups forms an extended series where H4 is extended as a compact hyperbolic and over-extended into a lorentzian group.
    The determinant of the Schläfli matrix by rank are:
    • det = 2n
    • det = n+1
    • det = 2
    • det = 4
    Determinants of the Schläfli matrix in exceptional series are:
    • det = 9-n, E4, E5, E6, E7 and E8, affine at E9, hyperbolic at E10)
    • det = 2
    • det = 3
    • det = 5-n to F4, affine at F5, hyperbolic at F6)
    • det = 3-n
    FiniteEn |
    Rank n,3n-7],3n-8]En=
    3
    E3=A2A1
    4
    A22

    A3A1

    E4=A4
    5
    B4A1

    D4A1

    A5

    A5

    E5=D5
    6
    A6

    B6

    D6

    E6

    B5A1

    D5A1

    D6

    E6 :File:DynkinE6Full.svg|*
    7]
    A6+=

    B6+=

    D6+=

    E6+=

    A7

    B7

    D7

    E7 :File:DynkinE7Full.svg|*

    E7 :File:DynkinE7Full.svg|*
    8,3]
    A6++=

    B6++=

    D6++=

    E6++=
    		]
    A7+= :File:AffineA7.svg|*

    B7+= :File:AffineB7.svg|*

    D7+= :File:AffineD7.svg|*

    E7+= :File:AffineE7.svg|*

    E8 :File:DynkinE8Full.svg|*
    9,3,3]
    A6+++

    B6+++

    D6+++

    E6+++
    		,3]
    A7++= :File:HyberbolicAffineA7.svg|*

    B7++= :File:HyberbolicAffineB7.svg|*

    D7++= :File:HyberbolicAffineD7.svg|*

    E7++= :File:HyberbolicAffineE7.svg|*

    E9=E8+= :File:E9-AffineE8.svg|*
    10,3,3]
    A7+++ :File:VeryExtendedAffineA7.svg|*

    B7+++ :File:VeryExtendedAffineB7.svg|*

    D7+++ :File:VeryExtendedAffineD7.svg|*

    E7+++ :File:VeryExtendedAffineE7.svg|*

    E10=E8++= :File:E10-HyperbolicAffineE8.svg|*
    11
    E11=E8+++ :File:E11-VeryExtendedAffineE8.svg|*
    Det724382429-n

    Geometric folding

    A Coxeter–Dynkin diagram that has a symmetry can be quotiented by the symmetry, yielding a new, generally multiply laced diagram, with the process called "folding".
    For example, in D4 folding to G2, the edge in G2 points from the class of the 3 outer nodes, to the class of the central node. And E8 folds into 2 copies of H4, the second copy scaled by τ.
    Geometrically this corresponds to orthogonal projections of uniform polytopes and tessellations. Notably, any finite simply-laced Coxeter–Dynkin diagram can be folded to I2, where h is the Coxeter number, which corresponds geometrically to a projection to the Coxeter plane.

    A few hyperbolic foldings

    Complex reflections

    Coxeter–Dynkin diagrams have been extended to complex space, Cn where nodes are unitary reflections of period greater than 2. Nodes are labeled by an index, assumed to be 2 for ordinary real reflection if suppressed. Coxeter writes the complex group, pr, as diagram.
    A 1-dimensional regular complex polytope in is represented as, having p vertices. Its real representation is a regular polygon,. Its symmetry is p or, order p. A unitary operator generator for is seen as a rotation in by 2π/p radians counter clockwise, and a edge is created by sequential applications of a single unitary reflection. A unitary reflection generator for a 1-polytope with p vertices is. When p = 2, the generator is eπi = –1, the same as a point reflection in the real plane.
    In a higher polytope, p or represents a p-edge element, with a 2-edge, or, representing an ordinary real edge between two vertices.

    Complex 1-polytopes,, represented in the Argand plane as regular polygons for p = 2, 3, 4, 5, and 6, with black vertices. The centroid of the p vertices is shown seen in red. The sides of the polygons represent one application of the symmetry generator, mapping each vertex to the next counterclockwise copy. These polygonal sides are not edge elements of the polytope, as a complex 1-polytope can have no edges and only contains vertex elements.


    12 irreducible Shephard groups with their subgroup index relations. Subgroups index 2 relate by removing a real reflection:
    p2 --> pp, index 2.
    pq --> pp, index q.
    p2 subgroups: p=2,3,4...
    p2 --> , index p
    p2 --> p×p, index 2

    Aa regular complex polygons in, has the form pr or Coxeter diagram. The symmetry group of a regular complex polygon is not called a Coxeter group, but instead a Shephard group, a type of Complex reflection group. The order of pr is.
    The rank 2 Shephard groups are: 22, p2, 33, 32, 33, 44, 32, 42, 43, 33, 55, 32, 52, and 53 or,,,,,,,,,,,,, of order 2q, 2p2, 24, 48, 72, 96, 144, 192, 288, 360, 600, 1200, and 1800 respectively.
    The symmetry group p1p2 is represented by 2 generators R1, R2, where: R1p1 = R2p2 = I. If q is even, q/2 = q/2. If q is odd, /2R2 = /2R1. When q is odd, p1=p2.
    The group or p is defined by 3 period 2 unitary reflections : R12 = R12 = R32 = 3 = 3 = 3 = p = 1. The period p can be seen as a double rotation in real.
    A similar group or is defined by 3 period 2 unitary reflections : R12 = R12 = R32 = 3 = 3 = 3 = p = 1.
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