Coxeter notation

For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram.

The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter diagram.

The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram.

Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like [...,3p,q] or [3p,q,r], starting with [31,1,1] or [3,31,1] = or as D4.

If the branch orders are equal, they can be grouped as an exponent as the length the cycle in brackets, like [(3,3,3,3)] = [3[4]], representing Coxeter diagram or .

The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs.

Coxeter's notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets, [X]+ which cuts the order of the group [X] in half, thus an index 2 subgroup.

The + operators can also be applied inside of the brackets, like [X,Y+] or [X,(Y,Z)+], and creates "semidirect" subgroups that may include both reflective and nonreflective generators.

Johnson extends the + operator to work with a placeholder 1+ nodes, which removes mirrors, doubling the size of the fundamental domain and cuts the group order in half.

The effect of a mirror removal is to duplicate connecting nodes, which can be seen in the Coxeter diagrams: = , or in bracket notation:[1+,2p, 1] = [1,p,1] = [p].

An odd-order adjacent branch, p, will not lower the group order, but create overlapping fundamental domains.

A Coxeter-Dynkin diagram can be marked up with explicit 2 branches defining a linear sequence of mirrors, open-nodes, and shared double-open nodes to show the chaining of the reflection generators.

Their common subgroup index 4 is [2+,2+], and is represented by (or ), with the double-open marking a shared node in the two alternations, and a single rotoreflection generator {012}.

The 4-dimensional double rotations, [2p+,2+,2q+] (with gcd(p,q)=1), which include a central group, and are expressed by Conway as ±[Cp×Cq],[5] order 2pq.

Half groups, [2p+,2+,2q+]+, or cyclic graph, [(2p+,2+,2q+,2+)], expressed by Conway is [Cp×Cq], order pq, with one generator, like {0123}.

In general a n-rotation group, [2p1+,2,2p2+,2,...,pn+] may require up to n generators if gcd(p1,..,pn)>1, as a product of all mirrors, and then swapping sequential pairs.

[6] For example, [4,4] has three independent nodes in the Coxeter diagram when the 4s are removed, so its commutator subgroup is index 23, and can have different representations, all with three + operators: [4+,4+]+, [1+,4,1+,4,1+], [1+,4,4,1+]+, or [(4+,4+,2+)].

Cyan, red, and green mirror lines correspond to the same colored nodes in the Coxeter diagram.

Subgroup generators can be expressed as products of the original 3 mirrors of the fundamental domain, {0,1,2}, corresponding to the 3 nodes of the Coxeter diagram, .

The same set of 15 small subgroups exists on all triangle groups with even order elements, like [6,4] in the hyperbolic plane: A parabolic subgroup of a Coxeter group can be identified by removing one or more generator mirrors represented with a Coxeter diagram.

But by considering the tetragonal disphenoid fundamental domain the [4] extended symmetry of the square graph can be marked more explicitly as [(2+,4)[3[4]]] or [2+,4[3[4]]].

An asterisk * superscript is effectively an inverse operation, creating radical subgroups removing connected of odd-ordered mirrors.

The + superscript simply implies that alternate mirror reflections are ignored, leaving the identity group in this simplest case.

The p-gonal subgroup [p]+, cyclic group Zp, of order p, generated by a rotation angle of π/p.

Coxeter notation uses double-bracking to represent an automorphic doubling of symmetry by adding a bisecting mirror to the fundamental domain.

When the + superscript is given inside of the brackets, it means reflections generated only from the adjacent mirrors (as defined by the Coxeter diagram, ) are alternated.

The polyhedral groups are based on the symmetry of platonic solids: the tetrahedron, octahedron, cube, icosahedron, and dodecahedron, with Schläfli symbols {3,3}, {3,4}, {4,3}, {3,5}, and {5,3} respectively.

In all these symmetries, alternate reflections can be removed producing the rotational tetrahedral [3,3]+(), octahedral [3,4]+ (), and icosahedral [3,5]+ () groups of order 12, 24, and 60.

Each node in the Coxeter diagram represents a mirror, by convention called ρi (and matrix Ri).

is unity, the transformation matrix can be expressed as: The reducible 3-dimensional finite reflective group is dihedral symmetry, [p,2], order 4p, .

A radical subgroup is similar to an alternation, but removes the rotational generators.
Rank 2 example, [6] trionic subgroups with 3 colors of mirror lines
Example on octahedral symmetry: [4,3 ⅄ ] = [2,4].
Example trionic subgroup on hexagonal symmetry [6,3] maps onto a larger [6,3] symmetry.
Rank 3
Example trionic subgroups on octagonal symmetry [8,3] maps onto larger [4,8] symmetries.
Rank 4
[3,3] ⅄ ≅ [2 + ,4] as one of 3 sets of 2 orthogonal mirrors in stereographic projection . The red, green, and blue represent 3 sets of mirrors, and the gray lines are removed mirrors, leaving 2-fold gyrations (purple diamonds).
Trionic relations of [3,3]
Trionic subgroup relations of [3,3,4]
A 2D central inversion is a 180 degree rotation, [2] +
Hasse diagram subgroups of [4,4], down to its commutator subgroup, index 8
A regular hexagon , with markings on edges and vertices has 8 symmetries: [6], [3], [2], [1], [6] + , [3] + , [2] + , [1] + , with [3] and [1] existing in two forms, depending whether the mirrors are on the edges or vertices.
Pyritohedral symmetry , [3+,4] is an index 5 subgroup of icosahedral symmetry , [5,3].
Hasse diagram with all subgroup relations on rank 2 Shephard groups.
Some subgroup relations among infinite Shephard groups
Example fundamental domains, [5,2], as spherical triangles
reflection lines for [3,3] =
Reflection lines for [4,3] =
Reflection lines for [5,3] =
[4,3,4] fundamental domain
[4,3 1,1 ] fundamental domain
[3 [4] ] fundamental domain