Coxeter–Dynkin diagram

Schläfli matrix

Every Coxeter diagram has a corresponding Schläfli matrix (so named after Ludwig Schläfli), A, with matrix elements a_i,j = a_j,i = −2 cos(π/p_i,j) where p_i,j is the branch order between mirrors i and j; that is, π/p_i,j is the dihedral angle between mirrors i and j. As a matrix of cosines, A is also called a Gramian matrix. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. A is closely related to the Cartan matrix, used in the similar but directed graph: the Dynkin diagram, in the limited cases of p = 2,3,4, and 6, which are generally not symmetric.

The determinant of the Schläfli matrix is called the Schläflian; the Schläflian and its sign determine whether the group is finite (positive), affine (zero), or indefinite (negative). This rule is called Schläfli's Criterion.

The eigenvalues of the Schläfli matrix determine 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 definitions:

• 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 its 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

The type of a rank 2 Coxeter group, i.e. generated by two different mirrors, is fully determined by the determinant of the Schläfli matrix, as this determinant is simply the product of the eigenvalues: finite (positive determinant), affine (zero determinant), or hyperbolic (negative determinant) type. Coxeter uses an equivalent bracket notation which lists sequences of branch orders as a substitute for the node-branch graphic diagrams. Rational solutions [p/q], , also exist, with gcd(p,q) = 1; these 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 spherical, Euclidean, or hyperbolic space of given dimension. (In 2D spaces, a mirror is a line; in 3D, a mirror is a plane.)

These visualizations show the fundamental domains for 2D and 3D Euclidean groups, and for 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; see footnote [a] below).

Application to 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 {p, q, r, ...}, 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 p, 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 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 2n-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.

Examples with 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 hosohedra 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 their dual 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, 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.

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 (positive) to affine (zero) to hyperbolic (negative), and ending as a Lorentzian group, containing at least one hyperbolic subgroup. The noncrystallographic H_n 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(A₁ⁿ = [2ⁿ⁻¹]) = 2ⁿ (finite for all n.)
• det(A_n = [3ⁿ⁻¹]) = n + 1 (finite for all n.)
• det(B_n = [4,3ⁿ⁻²]) = 2 (finite for all n.)
• det(D_n = [3^n−3,1,1]) = 4 (finite for all n.)

Determinants of the Schläfli matrix in exceptional series are:

• det(E_n = [3^n−3,2,1]) = 9 − n (finite for E₃ = A₂A₁, E₄ = A₄, E₅ = D₅, E₆, E₇ and E₈, affine at E₉ (${\displaystyle {\tilde{E}}_8}$), hyperbolic at E₁₀.)
• det([3^n−4,3,1]) = 2(8 − n) (finite for n = 4 to 7, affine (${\displaystyle {\tilde{E}}_7}$), and hyperbolic at n = 8.)
• det([3^n−4,2,2]) = 3(7 − n) (finite for n = 4 to 6, affine (${\displaystyle {\tilde{E}}_6}$), and hyperbolic at n = 7.)
• det(F_n = [3,4,3ⁿ⁻³]) = 5 − n (finite for F₃ = B₃ to F₄, affine at F₅ (${\displaystyle {\tilde{F}}_4}$), hyperbolic at F₆.)
• det(G_n = [6,3ⁿ⁻²]) = 3 − n (finite for G₂, affine at G₃ (${\displaystyle {\tilde{G}}_2}$), hyperbolic at G₄.)

Geometric folding

A (simply-laced) Coxeter–Dynkin diagram (finite, affine, or hyperbolic) that has a symmetry (satisfying one condition, below) 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 (valence 1), to the class of the central node (valence 3). 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(h), where h is the Coxeter number, which corresponds geometrically to a projection to the Coxeter plane.

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, p[q]r, as diagram .

A 1-dimensional regular complex polytope in ${\displaystyle {\mathbb{C}}^1}$ is represented as , having p vertices. Its real representation is a regular polygon, {p}. Its symmetry is _p[] or , order p. A unitary operator generator for is seen as a rotation in ${\displaystyle {\mathbb{R}}^2}$ 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 e^2πi/p = cos(2π/p) + i sin(2π/p). When p = 2, the generator is e^πi = −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.

A regular complex polygon in ${\displaystyle {\mathbb{C}}^2}$, has the form _p{q}_r 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 _p[q]_r is ${\displaystyle 8/q\cdot (1/p+2/q+1/r-1{)}^{-2}}$.

The rank 2 Shephard groups are: ₂[q]₂, _p[4]₂, ₃[3]₃, ₃[6]₂, ₃[4]₃, ₄[3]₄, ₃[8]₂, ₄[6]₂, ₄[4]₃, ₃[5]₃, ₅[3]₅, ₃[10]₂, ₅[6]₂, and ₅[4]₃ or , , , , , , , , , , , , , of order 2q, 2p², 24, 48, 72, 96, 144, 192, 288, 360, 600, 1200, and 1800 respectively.

The symmetry group _p1[q]_p2 is represented by 2 generators R₁, R₂, where:

R₁^p₁ = R₂^p₂ = I.

If q is even, (R₂R₁)^q/2 = (R₁R₂)^q/2. If q is odd, (R₂R₁)^(q-1)/2R₂ = (R₁R₂)^(q-1)/2R₁. When q is odd, p₁=p₂.

The ${\displaystyle {\mathbb{C}}^3}$ group or [1 1 1]^p is defined by 3 period 2 unitary reflections {R₁, R₂, R₃}:

R₁² = R₁² = R₃² = (R₁R₂)³ = (R₂R₃)³ = (R₃R₁)³ = (R₁R₂R₃R₁)^p = 1.

The period p can be seen as a double rotation in real ${\displaystyle {\mathbb{R}}^4}$.

A similar ${\displaystyle {\mathbb{C}}^3}$ group or [1 1 1]^(p) is defined by 3 period 2 unitary reflections {R₁, R₂, R₃}:

R₁² = R₁² = R₃² = (R₁R₂)³ = (R₂R₃)³ = (R₃R₁)³ = (R₁R₂R₃R₂)^p = 1.