Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (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 attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge), that is, the amount by which the angle between the reflective planes can be multiplied to get 180 degrees. An unlabeled branch implicitly represents order-3 (60 degrees), and each pair of nodes that is not connected by a branch at all (such as non-adjacent nodes) represents a pair of mirrors at order-2 (90 degrees).

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 (crystallographic) 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
(so named after Ludwig Schläfli), 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 (positive), affine (zero), indefinite (negative). This rule is called Schläfli's Criterion.

The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of ''finite type'' (all positive), ''affine type'' (all non-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 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 (i.e. have positive determinants), and paracompact if all its subgroups are finite or affine (i.e. have nonnegative determinants).

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 (or quasi-Lannér) 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 (positive determinant), affine type (zero determinant) or hyperbolic (negative determinant). Coxeter uses an equivalent bracket notation which lists sequences of branch orders as a substitute for the node-branch graphic diagrams. Rational solutions /q , also exist, with gcd(p,q)=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. (In 2D spaces, a mirror 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. For each the Coxeter diagram can be deduced by identifying the hyperplane mirrors and labelling their connectivity, ignoring 90-degree dihedral angles (order 2).

Finite Coxeter groups

:See also :polytope families for a table of end-node uniform polytopes associated with these groups.
*Three different symbols are given for the same groups – as a letter/number, as a bracketed set of numbers, and as the Coxeter diagram.
*The bifurcated D_''n'' group is a ''half'' or ''alternated'' version of the C_''n'' group.
*The bifurcated D_''n'' and E_''n'' groups are also labeled by a superscript form ^{''a'',''b'',''c''}where ''a'',''b'',''c'' are the numbers of segments in each of the three branches.

Application with uniform polytopes

Coxeter–Dynkin 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) 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(s) represented by the ringed node(s). (If two or more mirrors are marked, the vertex is equidistant from them.) A mirror is ''active'' (creates reflections) only with respect to points not on it. A diagram needs at least one active node to represent a polytope. An unconnected diagram (subgroups separated by order-2 branches, or orthogonal mirrors) 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 ''(k-1)''-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 (called a ''hole''). These shapes are alternations of polytopes with reflective symmetry, implying that every other vertex is deleted. The resulting polytope will have a subsymmetry of the 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 perpendicular to the mirror, represented as .
* Two unattached nodes represent two perpendicular mirrors. If both nodes are ringed, a rectangle can be created, 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 2''n''-gon if the point is off both mirrors. This forms the I₁(n) group.
* Two parallel mirrors can represent an infinite polygon I₁(∞) group, also called Ĩ₁.
* Three mirrors in a triangle form images seen in a traditional kaleidoscope and can be represented by three nodes 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.

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 (orthogonal) Coxeter groups like the uniform prisms, and can be seen more clearly as tilings of dihedrons and hosohedrons on the sphere, like this �[] or [6,2] family:

In comparison, the [6,3], 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 ,3 family produces a parallel set of uniform tilings, and their dual tilings. There is only 1 alternation (snub) 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. (Ernst Witt symbols for the affine groups are given as ''also'')
#$_$: diagrams of this type are cycles. (Also ''P''_''n'')
#$_$ is associated with the hypercube regular tessellation family. (Also ''R''_''n'')
#$_$ related to ''C'' by one removed mirror. (Also ''S''_''n'')
#$_$ related to C by two removed mirrors. (Also ''Q''_''n'')
#$_6$, $_7$, $_8$. (Also ''T''₇, ''T''₈, ''T''₉)
#$_4$ forms the regular tessellation. (Also ''U''₅)
#$_2$ forms 30-60-90 triangle fundamental domains. (Also ''V''₃)
#$_1$ is two parallel mirrors. ($=\tilde_1 = \tilde_1$) (Also ''W''₂)

Composite groups can also be defined as orthogonal projects. The most common use $\tilde_1$, like $\tilde_1^2$, represents square or rectangular checker board domains in the Euclidean plane. And $\tilde_1 \tilde_2$ represents triangular prism fundamental domains in Euclidean 3-space.

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 (Lannér simplices) exist as rank 3 to 5. Paracompact simplex groups (Koszul simplices) exist up to rank 10. Hypercompact (Vinberg polytopes) 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 (p q r) for:
:$\frac+\frac+\frac<1.$

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

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

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.

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 (a b c d ...), or equivalently in orbifold notation as *''abcd''.... Coxeter–Dynkin diagrams for these polygonal kaleidoscopes can be seen as a degenerate (n-1)-simplex fundamental domains, with a cyclic of branches order a,b,c... and the remaining n*(n-3)/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 , infin;,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 , infin;,3,∞or π/λ₁,3,iπ/λ₂(orbifold *3222), where (λ₁,λ₂) 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 with 4 perimeter branches as ideal vertices and two diagonal branches as infinity (shown as dotted lines) 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: /2,5,3,3= and ,5/2,5,3= .

Ranks 4–5

The fundamental domain of either of the two bifurcating groups, ,3^1,1and ,3,3^1,1 is double that of a corresponding linear group, ,3,4and ,3,3,4respectively. Letter names are given by Johnson as extended Witt symbols.

Paracompact (Koszul simplex groups)

Paracompact (also called noncompact) 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 (integer ratio of volumes) 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 /''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 ''H''^''n'' 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 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 infin;,3,3,∞which can be constructed by a mirror removal of paracompact hyperbolic group ,4,4as ,4,1⁺,4 The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramids include ,4,1⁺,4= infin;,4,4,∞ = . Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow-tie graphs: 3,3,4,1⁺,4)= (3,∞,3)),((3,∞,3))or , 3,4,4,1⁺,4)= (4,∞,3)),((3,∞,4))or , 4,4,4,1⁺,4)= (4,∞,4)),((4,∞,4))or .

Other valid paracompact graphs with quadrilateral pyramid fundamental domains include:

Another subgroup ⁺,4^1,1,1= infin;,4,1⁺,4,∞= infin;^[6/sup>">.html" ;"title="infin;^{[6">infin;[6/sup> = = .}

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 7-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 manifold#Lorentzian manifold">Lorentzian geometry