Coxeter notation

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Fundamental domains of reflective 3D point groups
CDel node.png, [ ]=[1]
C1v
CDel node.pngCDel 2.pngCDel node.png, [2]
C2v
CDel node.pngCDel 3.pngCDel node.png, [3]
C3v
CDel node.pngCDel 4.pngCDel node.png, [4]
C4v
CDel node.pngCDel 5.pngCDel node.png, [5]
C5v
CDel node.pngCDel 6.pngCDel node.png, [6]
C6v
Spherical digonal hosohedron.png
Order 2
Spherical square hosohedron.png
Order 4
Spherical hexagonal hosohedron.png
Order 6
Spherical octagonal hosohedron.png
Order 8
Spherical decagonal hosohedron.png
Order 10
Spherical dodecagonal hosohedron.png
Order 12
CDel node.pngCDel 2.pngCDel node.png
[2]=[2,1]
D1h
CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
[2,2]
D2h
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
[2,3]
D3h
CDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png
[2,4]
D4h
CDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png
[2,5]
D5h
CDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png
[2,6]
D6h
Spherical digonal bipyramid.png
Order 4
Spherical square bipyramid.png
Order 8
Spherical hexagonal bipyramid.png
Order 12
Spherical octagonal bipyramid.png
Order 16
Spherical decagonal bipyramid.png
Order 20
Spherical dodecagonal bipyramid.png
Order 24
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, [3,3], Td CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, [4,3], Oh CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png, [5,3], Ih
Spherical tetrakis hexahedron-3edge-color.png
Order 24
Spherical disdyakis dodecahedron-3and1-color.png
Order 48
Spherical compound of five octahedra.png
Order 120
Coxeter notation expresses Coxeter groups as a list of branch orders of a Coxeter diagram, like the polyhedral groups, CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png = [p,q]. dihedral groups, CDel node.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png, can be expressed a product [ ]×[n] or in a single symbol with an explicit order 2 branch, [2,n].

In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

Reflectional groups[edit]

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. It uses the same simplification, suppressing 2s between orthogonal mirrors.

The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the An group is represented by [3n-1], to imply n nodes connected by n-1 order-3 branches. Example A2 = [3,3] = [32] or [31,1] represents diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png or CDel node.pngCDel split1.pngCDel nodes.png.

Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like [3p,q,r], starting with [31,1,1] = CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png as D4. Coxeter allowed for zeros as special cases to fit the An family, like A3 = [3,3,3,3] = [34,0,0] = [33,1,0] = [32,2,0], like CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

Coxeter groups formed by cyclic diagrams are represented by parenthesese inside of brackets, like [(p,q,r)] = CDel pqr.png for the triangle group (p q r). 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 CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png or CDel branch.pngCDel 3ab.pngCDel branch.png. CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.png can be represented as [3,(3,3,3)] or [3,3[3]].

More complicated looping diagrams can also be expressed with care. The paracompact Coxeter group CDel node.pngCDel split1.pngCDel branch.pngCDel split2.pngCDel node.png can be represented by Coxeter notation [(3,3,(3),3,3)], with nested/overlapping parentheses showing two adjacent [(3,3,3)] loops, and is also represented more compactly as [3[ ]×[ ]], representing the rhombic symmetry of the Coxeter diagram. The paracompact complete graph diagram CDel tet.png or CDel branch.pngCDel splitcross.pngCDel branch.png, is represented as [3[3,3]] with the superscript [3,3] as the symmetry of its regular tetrahedron coxeter diagram.

The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs. So the Coxeter diagram CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png = A2×A2 = 2A2 can be represented by [3]×[3] = [3]2 = [3,2,3].

Finite groups
Rank Group
symbol
Bracket
notation
Coxeter
diagram
2 A2 [3] CDel node.pngCDel 3.pngCDel node.png
2 B2 [4] CDel node.pngCDel 4.pngCDel node.png
2 H2 [5] CDel node.pngCDel 5.pngCDel node.png
2 G2 [6] CDel node.pngCDel 6.pngCDel node.png
2 I2(p) [p] CDel node.pngCDel p.pngCDel node.png
3 Ih, H3 [5,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
3 Td, A3 [3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3 Oh, B3 [4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
4 A4 [3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4 B4 [4,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4 D4 [31,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
4 F4 [3,4,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
4 H4 [5,3,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
n An [3n-1] CDel node.pngCDel 3.pngCDel node.pngCDel 3.png..CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
n Bn [4,3n-2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
n Dn [3n-3,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6 E6 [32,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
7 E7 [33,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
8 E8 [34,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Affine groups
Group
symbol
Bracket
notation
Coxeter diagram
[∞] CDel node.pngCDel infin.pngCDel node.png
[3[3]] CDel node.pngCDel split1.pngCDel branch.png
[4,4] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
[6,3] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
[3[4]] CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png
[4,31,1] CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
[4,3,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[3[5]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
[4,3,31,1] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[4,3,3,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[ 31,1,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png
[3,4,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3[n+1]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.png...CDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
or
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.png...CDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
[4,3n-3,31,1] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[4,3n-2,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[ 31,1,3n-4,31,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[32,2,2] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branchbranch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
[33,3,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
[35,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Hyperbolic groups
Group
symbol
Bracket
notation
Coxeter
diagram
[p,q]
with 2(p+q)<pq
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
[(p,q,r)]
with
CDel pqr.png
[4,3,5] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[5,3,5] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[3,5,3] CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
[5,31,1] CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
[(3,3,3,4)] CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png 
[(3,3,3,5)] CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png 
[(3,4,3,4)] CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
[(3,4,3,5)] CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
[(3,5,3,5)] CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
[3,3,3,5] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[4,3,3,5] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[5,3,3,5] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[5,3,31,1] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[(3,3,3,3,4)] CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's diagram.

Subgroups[edit]

Coxeter's notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets which cuts the order of the group in half (called index 2 subgroup). This is called a direct subgroup because what remains are only direct isometries without reflective symmetry.

+ operators can also be applied inside of the brackets, and creates "semidirect" subgroups that include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches next to it. Elements by parentheses inside of a Coxeter group can be give a + superscript operator, having the effect of dividing adjacent ordered branches into half order, thus is usually only applied with even numbers. For example, [4,3+] (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png) and [4,(3,3)+] (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png). The subgroup index is 2n for n + operators.

Groups without neighboring + elements can be seen in ringed nodes Coxeter-Dynkin diagram for uniform polytopes and honeycomb are related to hole nodes around the + elements, empty circles with the alternated nodes removed. So the snub cube, CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png has symmetry [4,3]+ (CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png), and the snub tetrahedron, CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png has symmetry [4,3+] (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png), and a demicube, h{4,3} = {3,3} (CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png or CDel nodes 10ru.pngCDel split2.pngCDel node.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png) has symmetry [1+,4,3] = [3,3] (CDel node h2.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png or CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel nodes.pngCDel split2.pngCDel node.png = CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png).

Halving subgroups[edit]

Example halving operations
Dihedral symmetry domains 4.png Dihedral symmetry 4 half1.png
CDel node c1.pngCDel 4.pngCDel node c3.png
[ 1,4, 1] = [4]
CDel node h0.pngCDel 4.pngCDel node c3.png = CDel node c3.pngCDel 2x.pngCDel node c3.png = CDel node c3.pngCDel 2.pngCDel node c3.png
[1+,4, 1]=[2]=[ ]×[ ]
Dihedral symmetry 4 half2.png Cyclic symmetry 4 half.png
CDel node c1.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel 2x.pngCDel node c1.png = CDel node c1.pngCDel 2.pngCDel node c1.png
[ 1,4,1+]=[2]=[ ]×[ ]
CDel node h0.pngCDel 4.pngCDel node h0.png = CDel node h0.pngCDel 4.pngCDel node h2.png = CDel node h2.pngCDel 4.pngCDel node h0.png = CDel node h2.pngCDel 2x.pngCDel node h2.png
[1+,4,1+] = [2]+

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. In general this operation only applies to mirrors bounded by all even-order branches. The 1 represents a mirror so [2p] can be seen as [2p,1], [1,2p], or [1,2p,1], like diagram CDel node.pngCDel 2x.pngCDel p.pngCDel node.png or CDel node c1.pngCDel 2x.pngCDel p.pngCDel node c3.png, with 2 mirrors related by an order-2p dihedral angle. The effect of a mirror removal is to duplicate connecting nodes, which can be seen in the Coxeter diagrams: CDel node h0.pngCDel 2x.pngCDel p.pngCDel node c3.png = CDel labelp.pngCDel branch c3.png, or in bracket notation:[1+,2p, 1] = [1,p,1] = [p].

Each of these mirrors can be removed so h[2p] = [1+,2p,1] = [1,2p,1+] = [p], a reflective subgroup index 2. This can be shown in a Coxeter diagram by adding a + symbol above the node: CDel node h0.pngCDel 2x.pngCDel p.pngCDel node.png = CDel node.pngCDel 2x.pngCDel p.pngCDel node h0.png = CDel labelp.pngCDel branch.png.

If both mirrors are removed, a quarter subgroup is generated, with the branch order becoming a gyration point of half the order:

q[2p] = [1+,2p,1+] = [p]+, a rotational subgroup of index 4. CDel node h0.pngCDel 2x.pngCDel p.pngCDel node h0.png = CDel node h0.pngCDel 2x.pngCDel p.pngCDel node h2.png = CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h0.png = CDel labelp.pngCDel branch h2h2.png.

For example, (with p=2): [4,1+] = [1+,4] = [2] = [ ]×[ ], order 4. [1+,4,1+] = [2]+, order 2.

The opposite to halving is doubling which adds a mirror, bisecting a fundamental domain, and doubling the group order.

[[p]] = [2p]

Halving operations apply for higher rank groups, like h[4,3] = [1+,4,3] = [3,3], removing half the mirrors at the 4-branch. The effect of a mirror removal is to duplicate all connecting nodes, which can be seen in the Coxeter diagrams: CDel node h0.pngCDel 2x.pngCDel p.pngCDel node c1.pngCDel 3.pngCDel node c2.png = CDel labelp.pngCDel branch c1.pngCDel split2.pngCDel node c2.png, h[2p,3] = [1+,2p,3] = [(p,3,3)].

Doubling by adding a mirror also applies in reversing the halving operation: [[3,3]] = [4,3], or more generally [[(q,q,p)]] = [2p,q].

Tetrahedral symmetry Octahedral symmetry
Sphere symmetry group td.png
Td, [3,3] = [1+,4,3]
CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png = CDel nodeab c1.pngCDel split2.pngCDel node c1.png = CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png
(Order 24)
Sphere symmetry group oh.png
Oh, [4,3] = [[3,3]]
CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png
(Order 48)

Radical subgroups[edit]

A radical subgroup is similar to an alternation, but removes the rotational generators.

Johnson also added an asterisk or star * operator, that acts similar to the + operator, but removes rotational symmetry. The index of the radical subgroup is the order of the removed element. For example, [4,3*] ≅ [2,2]. The removed [3] subgroup is order 6 so [2,2] is an index 6 subgroup of [4,3].

The radical subgroups represent the inverse operation to an extended symmetry operation. For example, [4,3*] ≅ [2,2], and in reverse [2,2] can be extended as [3[2,2]] ≅ [4,3]. The subgroups can be expressed as a Coxeter diagram: CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.png. The removed node (mirror) causes adjacent mirror virtual mirrors to become real mirrors.

If [4,3] has generators {0,1,2}, [4,3+], index 2, has generators {0,12}; [1+,4,3], index 2 has generators {010,1,2}; while radical subgroup [4,3*], index 6, has generators {01210, 2, (012)3}; and finally [1+,4,3*], index 12 has generators {0(12)20, (012)201}.

Trionic subgroups[edit]

[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]

Johnson identified two specific subgroups of [3,3], first an index 3 subgroup [3,3] ≅ [2+,4], with [3,3] (CDel node n0.pngCDel 3.pngCDel node n1.pngCDel 3.pngCDel node n2.png = CDel node.pngCDel split1.pngCDel nodes.png) generators {0,1,2}. It can also be written as [(3,3,2)] (CDel node.pngCDel split1.pngCDel 2.pngCDel branch h2h2.pngCDel label2.png) as a reminder of its generators {02,1}. This symmetry reduction is the relationship between the regular tetrahedron and the tetragonal disphenoid, represent a stretching of a tetrahedron perpendicular to two opposite edges.

Secondly he identifies a related index 6 subgroup [3,3]Δ or [(3,3,2)]+, index 3 from [3,3]+ ≅ [2,2]+, with generators {02,1021}, from [3,3] and its generators {0,1,2}.

These subgroups also apply within larger Coxeter groups with [3,3] subgroup with neighboring branches all even order.

Trionic subgroup relations of [3,3,4]

For example, [(3,3)+,4], [(3,3),4], and [(3,3)Δ,4] are subgroups of [3,3,4], index 2, 3 and 6 respectively. The generators of [(3,3),4] ≅ [[4,2,4]] ≅ [8,2+,8], order 128, are {02,1,3} from [3,3,4] generators {0,1,2,3}. And [(3,3)Δ,4] ≅ [[4,2+,4]], order 64, has generators {02,1021,3}.

Also related [31,1,1] = [3,3,4,1+] has trionic subgroups: [31,1,1] = [(3,3),4,1+], order 64, and [31,1,1]Δ = [(3,3)Δ,4,1+] ≅ [[4,2+,4]]+, order 32.

Central inversion[edit]

A 2D central inversion is a 180 degree rotation, [2]+

A central inversion, order 2, is operationally differently by dimension. The group [ ]n = [2n-1] represents n orthogonal mirrors in n-dimensional space, or an n-flat subspace of a higher dimensional space. The mirrors of the group [2n-1] are numbered 0..n-1. The order of the mirrors doesn't matter in the case of an inversion.

From that basis, the central inversion has a generator as the product of all the orthogonal mirrors. In Coxeter notation this inversion group is expressed by adding an alternation + to each 2 branch. The alternation symmetry is marked on Coxeter diagram nodes as open nodes.

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.

For example, [2+,2] and [2,2+] are subgroups index 2 of [2,2], CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png, and are represented as CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.png and CDel node.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h2.png with generators {01,2} and {0,12} respectively. Their common subgroup index 4 is [2+,2+], and is represented by CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png, with the double-open CDel node h4.png marking a shared node in the two alternations, and a single rotoreflection generator {012}.

Dimension Coxeter notation Order Coxeter diagram Operation Generator
2 [2]+ 2 CDel node h2.pngCDel 2x.pngCDel node h2.png 180° rotation, C2 {01}
3 [2+,2+] 2 CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png rotoreflection, Ci or S2 {012}
4 [2+,2+,2+] 2 CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png double rotation {0123}
5 [2+,2+,2+,2+] 2 CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png double rotary reflection {01234}
6 [2+,2+,2+,2+,2+] 2 CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png triple rotation {012345}
7 [2+,2+,2+,2+,2+,2+] 2 CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png triple rotary reflection {0123456}

Rotations and rotary reflections[edit]

Rotations and rotary reflections are constructed by a single single-generator product of all the reflections of a prismatic group, [2p]×[2q]×... When gcd(p,q,..)=1, they are isomorphic to the abstract cyclic group Zn, of order n=2pq.

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

Dimension Coxeter notation Order Coxeter diagram Operation Generator Direct subgroup
2 [p]+ p CDel node h2.pngCDel p.pngCDel node h2.png Rotation {01} [p]+
3 [2p+,2+] 2p CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h2.png rotary reflection {012}
4 [2p+,2+,2+] CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png double rotation {0123}
5 [2p+,2+,2+,2+] CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png double rotary reflection {01234}
6 [2p+,2+,2+,2+,2+] CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png triple rotation {012345}
7 [2p+,2+,2+,2+,2+,2+] CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png triple rotary reflection {0123456}
4 [2p+,2+,2q+] 2pq CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h2.png double rotation {0123} [p+,2,q+]
5 [2p+,2+,2q+,2+] CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.pngCDel node h2.png double rotary reflection {01234}
6 [2p+,2+,2q+,2+,2+] CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png triple rotation {012345}
7 [2p+,2+,2q+,2+,2+,2+] CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png triple rotary reflection {0123456}
6 [2p+,2+,2q+,2+,2r+] 2pqr CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel r.pngCDel node h2.png triple rotation {012345} [p+,2,q+,2,r+]
7 [2p+,2+,2q+,2+,2r+,2+] CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel r.pngCDel node h4.pngCDel 2x.pngCDel node h2.png triple rotary reflection {0123456}

Commutator subgroups[edit]

Simple groups with only odd-order branch elements have only a single rotational/translational subgroup of order 2, which is also the commutator subgroup, examples [3,3]+, [3,5]+, [3,3,3]+, [3,3,5]+. For other Coxeter groups with even-order branches, the commutator subgroup has index 2c, where c is the number of disconnected subgraphs when all the even-order branches are removed.[2] 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+)]. A general notation can be used with +c as a group exponent, like [4,4]+3.

Example subgroups[edit]

Rank 2 example subgroups[edit]

Dihedral symmetry groups with even-orders have a number of subgroups. This example shows two generator mirrors of [4] in red and green, and looks at all subgroups by halfing, rank-reduction, and their direct subgroups. The group [4], CDel node n0.pngCDel 4.pngCDel node n1.png has two mirror generators 0, and 1. Each generate two virtual mirrors 101 and 010 by reflection across the other.

Rank 3 Euclidean example subgroups[edit]

The [4,4] group has 15 small index subgroups. This table shows them all, with a yellow fundamental domain for pure reflective groups, and alternating white and blue domains which are paired up to make rotational domains. 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, CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 4.pngCDel node n2.png. A product of two intersecting reflection lines makes a rotation, like {012}, {12}, or {02}. Removing a mirror causes two copies of neighboring mirrors, across the removed mirror, like {010}, and {212}. Two rotations in series cut the rotation order in half, like {0101} or {(01)2}, {1212} or {(02)2}. A product of all three mirrors creates a transreflection, like {012} or {120}.

Hyperbolic example subgroups[edit]

The same set of 15 small subgroups exists on all triangle groups with even order elements, like [6,4] in the hyperbolic plane:

Extended symmetry[edit]

Wallpaper
group
Triangle
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
p3m1 (*333) a1 Triangle symmetry1.png [3[3]] CDel node.pngCDel split1.pngCDel branch.png (none)
p6m (*632) i2 Triangle symmetry3.png [[3[3]]] ↔ [6,3] CDel node c1.pngCDel split1.pngCDel branch c2.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 6.pngCDel node.png ×2 ↔ CDel node 1.pngCDel split1.pngCDel branch.png 1, CDel node.pngCDel split1.pngCDel branch 11.png 2
p31m (3*3) g3 Triangle symmetry2.png [3+[3[3]]] ↔ [6,3+] ×3 ↔ ½ (none)
p6 (632) r6 Triangle symmetry4.png [3[3[3]]]+ ↔ [6,3]+ CDel node c1.pngCDel split1.pngCDel branch c1.pngCDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png ½×6 ↔ ½ CDel node h.pngCDel split1.pngCDel branch hh.png (1)
p6m (*632) [3[3[3]]] ↔ [6,3] ×6 ↔ CDel node 1.pngCDel split1.pngCDel branch 11.png 3
In the Euclidean plane, the , [3[3]] Coxeter group can be extended in two ways into the , [6,3] Coxeter group and relates uniform tilings as ringed diagrams.

Coxeter's notation includes double square bracket notation, [[X]] to express automorphic symmetry within a Coxeter diagram. Johnson added alternative of angled-bracket <[X]> option as equivalent to square brackets for doubling to distinguish diagram symmetry through the nodes versus through the branches. Johnson also added a prefix symmetry modifier [Y[X]], where Y can either represent symmetry of the Coxeter diagram of [X], or symmetry of the fundamental domain of [X].

For example, in 3D these equivalent rectangle and rhombic geometry diagrams of : CDel branch.pngCDel 3ab.pngCDel 3ab.pngCDel branch.png and CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png, the first doubled with square brackets, [[3[4]]] or twice doubled as [2[3[4]]], with [2], order 4 higher symmetry. To differentiate the second, angled brackets are used for doubling, <[3[4]]> and twice doubled as <2[3[4]]>, also with a different [2], order 4 symmetry. Finally a full symmetry where all 4 nodes are equivalent can be represented by [4[3[4]]], with the order 8, [4] symmetry of the square. 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]]].

Further symmetry exists in the cyclic and branching , , and diagrams. has order 2n symmetry of a regular n-gon, {n}, and is represented by [n[3[n]]]. and are represented by [3[31,1,1]] = [3,4,3] and [3[32,2,2]] respectively while by [(3,3)[31,1,1,1]] = [3,3,4,3], with the diagram containing the order 24 symmetry of the regular tetrahedron, {3,3}. The paracompact hyperbolic group = [31,1,1,1,1], CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel split1.pngCDel nodes.png, contains the symmetry of a 5-cell, {3,3,3}, and thus is represented by [(3,3,3)[31,1,1,1,1]] = [3,4,3,3,3].

An asterisk * superscript is effectively an inverse operation, creating radical subgroups removing connected of odd-ordered mirrors.[3]

Examples:

Example Extended groups and radical subgroups
Extended groups Radical subgroups Coxeter diagrams Index
[3[2,2]] = [4,3] [4,3*] = [2,2] CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel node c1.pngCDel 2.pngCDel nodeab c1.png 6
[(3,3)[2,2,2]] = [4,3,3] [4,(3,3)*] = [2,2,2] CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png = CDel nodeab c1.pngCDel 2.pngCDel nodeab c1.png 24
[1[31,1]] = [[3,3]] = [3,4] [3,4,1+] = [3,3] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel split1.pngCDel nodeab c2.png 2
[3[31,1,1]] = [3,4,3] [3*,4,3] = [31,1,1] CDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.png = CDel node c1.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.png 6
[2[31,1,1,1]] = [4,3,3,4] [1+,4,3,3,4,1+] = [31,1,1,1] CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node h0.png = CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel split1.pngCDel nodeab c3.png 4
[3[3,31,1,1]] = [3,3,4,3] [3*,4,3,3] = [31,1,1,1] CDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png = CDel node c1.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.pngCDel 3.pngCDel node c3.png 6
[(3,3)[31,1,1,1]] = [3,4,3,3] [3,4,(3,3)*] = [31,1,1,1] CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png = CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png 24
[2[3,31,1,1,1]] = [3,(3,4)1,1] [3,(3,4,1+)1,1] = [3,31,1,1,1] CDel node c4.pngCDel 3.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel 4a4b.pngCDel nodes.png = CDel node c4.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c3.pngCDel split1.pngCDel nodeab c2.png 4
[(2,3)[1,131,1,1]] = [4,3,3,4,3] [1+,4,3,3,4,3+] = [31,1,1,1,1] CDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.pngCDel split1.pngCDel nodeab c3.png 12
[(3,3)[3,31,1,1,1]] = [3,3,4,3,3] [3,3,4,(3,3)*] = [31,1,1,1,1] CDel node c3.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png = CDel node c3.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png 24
[(3,3,3)[31,1,1,1,1]] = [3,4,3,3,3] [3,4,(3,3,3)*] = [31,1,1,1,1] CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node g.pngCDel 3g.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png = CDel node c1.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png 120
Extended groups Radical subgroups Coxeter diagrams Index
[1[3[3]]] = [3,6] [3,6,1+] = [3[3]] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 6.pngCDel node h0.png = CDel node c1.pngCDel split1.pngCDel branch c2.png 2
[3[3[3]]] = [6,3] [6,3*] = [3[3]] CDel node c1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel node c1.pngCDel split1.pngCDel branch c1.png 6
[1[3,3[3]]] = [3,3,6] [3,3,6,1+] = [3,3[3]] CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 6.pngCDel node h0.png = CDel node c1.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel branch c3.png 2
[(3,3)[3[3,3]]] = [6,3,3] [6,(3,3)*] = [3[3,3]] CDel node c1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png = CDel node c1.pngCDel splitsplit1.pngCDel branch4 c1.pngCDel splitsplit2.pngCDel node c1.png 24
[1[∞]2] = [4,4] [4,1+,4] = [∞]2 = [∞]×[∞] = [∞,2,∞] CDel node c1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node c2.png = CDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel branch c1-2.pngCDel labelinfin.png 2
[2[∞]2] = [4,4] [1+,4,4,1+] = [(4,4,2*)] = [∞]2 CDel node h0.pngCDel 4.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel labelinfin.pngCDel branch c2.pngCDel 2.pngCDel branch c2.pngCDel labelinfin.png 4
[4[∞]2] = [4,4] [4,4*] = [∞]2 CDel node c1.pngCDel 4.pngCDel node g.pngCDel 4sg.pngCDel node g.png = CDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel branch c1.pngCDel labelinfin.png 8
[2[3[4]]] = [4,3,4] [1+,4,3,4,1+] = [(4,3,4,2*)] = [3[4]] CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel split2.pngCDel node c1.png = CDel nodeab c1.pngCDel splitcross.pngCDel nodeab c2.png 4
[3[∞]3] = [4,3,4] [4,3*,4] = [∞]3 = [∞,2,∞,2,∞] CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node c2.png = CDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel labelinfin.pngCDel branch c1-2.png 6
[(3,3)[∞]3] = [4,31,1] [4,(31,1)*] = [∞]3 CDel node c1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.png 24
[(4,3)[∞]3] = [4,3,4] [4,(3,4)*] = [∞]3 CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4g.pngCDel node g.png = CDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.png 48
[(3,3)[∞]4] = [4,3,3,4] [4,(3,3)*,4] = [∞]4 CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.pngCDel 4.pngCDel node c2.png = CDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel labelinfin.pngCDel branch c1-2.png 24
[(4,3,3)[∞]4] = [4,3,3,4] [4,(3,3,4)*] = [∞]4 CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3g.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4g.pngCDel node g.png = CDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.png 384

Looking at generators, the double symmetry is seen as adding a new operator that maps symmetric positions in the Coxeter diagram, making some original generators redundant. For 3D space groups, and 4D point groups, Coxeter defines an index two subgroup of [[X]], [[X]+], which he defines as the product of the original generators of [X] by the doubling generator. This looks similar to [[X]]+, which is the chiral subgroup of [[X]]. So for example the 3D space groups [[4,3,4]]+ (I432, 211) and [[4,3,4]+] (Pm3n, 223) are distinct subgroups of [[4,3,4]] (Im3m, 229).

Computation with reflection matrices as symmetry generators[edit]

A Coxeter group, represented by Coxeter diagram CDel node n0.pngCDel p.pngCDel node n1.pngCDel q.pngCDel node n2.png, is given Coxeter notation [p,q] for the branch orders. Each node in the Coxeter diagram represents a mirror, by convention called ρi (and matrix Ri). The generators of this group [p,q] are reflections: ρ0, ρ1, and ρ2. Rotational subsymmetry is given as products of reflections: By convention, σ0,1 (and matrix S0,1) = ρ0ρ1 represents a rotation of angle π/p, and σ1,2 = ρ1ρ2 is a rotation of angle π/q, and σ0,2 = ρ0ρ2 represents a rotation of angle π/2.

[p,q]+ is an index 2 subgroup represented by two rotation generators, each a products of two reflections: σ0,1, σ1,2, and representing rotations of π/p, and π/q angles respectively.

If q is even, [p+,q] is another subgroup of index 2, represented by rotation generator σ0,1, and reflectional ρ2.

If both p and q are even, [p+,q+] is a subgroup of index 4 with two generators, constructed as a product of all three reflection matrices: By convention as: ψ0,1,2 and ψ1,2,0, which are rotary reflections, representing a reflection and rotation or reflection.

In the case of affine Coxeter groups like CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 4.pngCDel node n2.png, or CDel node n0.pngCDel infin.pngCDel node n1.png, one mirror, usually the last, is translated off the origin. A translation generator τ0,1 (and matrix T0,1) is constructed as the product of two (or an even number of) reflections, including the affine reflection. A transreflection (reflection plus a translation) can be the product of an odd number of reflections φ0,1,2 (and matrix V0,1,2), like the index 4 subgroup CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 4.pngCDel node n2.png: [4+,4+] = CDel node h2.pngCDel 4.pngCDel node h4.pngCDel 4.pngCDel node h2.png.

Another composite generator, by convention as ζ (and matrix Z), represents the inversion, mapping a point to its inverse. For [4,3] and [5,3], ζ = (ρ0ρ1ρ2)h/2, where h is 6 and 10 respectively, the Coxeter number for each family. For 3D Coxeter group [p,q] (CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png), this subgroup is a rotary reflection [2+,h+].

Coxeter groups are categorized by their rank, being the number of nodes in its Coxeter-Dynkin diagram. The structure of the groups are also given with their abstract group types: In this article, the abstract dihedral groups are represented as Dihn, and cyclic groups are represented by Zn, with Dih1=Z2.

Rank 2[edit]

Example, in 2D, the Coxeter group [p] (CDel node.pngCDel p.pngCDel node.png) is represented by two reflection matrices R0 and R1, The cyclic symmetry [p]+ (CDel node h2.pngCDel p.pngCDel node h2.png) is represented by rotation generator of matrix S0,1.

[p], CDel node n0.pngCDel p.pngCDel node n1.png
Reflections Rotation
R0 R1 S0,1=R0×R1

[2], CDel node n0.pngCDel 2.pngCDel node n1.png
Reflections Rotation
R0 R1 S0,1=R0×R1

[4], CDel node n0.pngCDel 4.pngCDel node n1.png
Reflections Rotation
R0 R1 S0,1=R0×R1

Rank 3[edit]

The simplest irreducible 3-dimensional finite reflective group is [3,3], order 24, CDel node n0.pngCDel 3.pngCDel node n1.pngCDel 3.pngCDel node n2.png. The reflection generators, from a D3=A3 construction, are matrices R0, R1, R2. R02=R12=R22=(R0×R1)3=(R1×R2)3=(R0×R2)2=Identity. [3,3]+ (CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png) is generated by 2 of 3 rotations: S0,1, S1,2, and S0,2. A trionic subgroup, isomorphic to [2+,4], is generated by S0,2 and R1. A 4-fold rotoreflection is generated by V0,1,2.

[3,3], CDel node n0.pngCDel 3.pngCDel node n1.pngCDel 3.pngCDel node n2.png
Reflections Rotations Rotoreflection
R0 R1 R2 S0,1=R0×R1 S1,2=R1×R2 S0,2=R0×R2 V0,1,2=R0×R1×R2

Another irreducible 3-dimensional finite reflective group is [4,3], order 48, CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 3.pngCDel node n2.png. The reflection generators matrices are R0, R1, R2. R02=R12=R22=(R0×R1)4=(R1×R2)3=(R0×R2)2=Identity. [4,3]+ (CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png) is generated by 2 of 3 rotations: S0,1, S1,2, and S0,2. A 6-fold rotoreflection is generated by V0,1,2.

[4,3], CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 3.pngCDel node n2.png
Reflections Rotations Rotoreflection
R0 R1 R2 S0,1 S1,2 S0,2 V0,1,2

Affine rank 3[edit]

A simple example affine group is [4,4] (CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 4.pngCDel node n2.png) (p4m), can be given by three reflection matrices, constructed as a reflection across the x axis (y=0), a diagonal (x=y), and the affine reflection across the line (x=1). [4,4]+ (CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h2.png) (p4) is generated by S0,1 S1,2, and S0,2. [4+,4+] (CDel node h2.pngCDel 4.pngCDel node h4.pngCDel 4.pngCDel node h2.png) (pgg) is generated by 2-fold rotation S0,2 and transreflection V0,1,2. [4+,4] (CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node.png) (p4g) is generated by S0,1 and R3. The group [(4,4,2+)] (CDel node.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png) (cmm), is generated by 2-fold rotation S1,3 and reflection R2.

[4,4], CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 4.pngCDel node n2.png
Reflections Rotations Rotoreflection
R0 R1 R2 S0,1 S1,2 S0,2 V0,1,2

Rank one groups[edit]

In one dimension, the bilateral group [ ] represents a single mirror symmetry, abstract Dih1 or Z2, symmetry order 2. It is represented as a Coxeter–Dynkin diagram with a single node, CDel node.png. The identity group is the direct subgroup [ ]+, Z1, symmetry order 1. The + superscript simply implies that alternate mirror reflections are ignored, leaving the identity group in this simplest case. Coxeter used a single open node to represent an alternation, CDel node h2.png.

Group Coxeter notation Coxeter diagram Order Description
C1 [ ]+ CDel node h2.png 1 Identity
D1 [ ] CDel node.png 2 Reflection group

Rank two groups[edit]

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.

In two dimensions, the rectangular group [2], abstract D12 or D2, also can be represented as a direct product [ ]×[ ], being the product of two bilateral groups, represents two orthogonal mirrors, with Coxeter diagram, CDel node.pngCDel 2.pngCDel node.png, with order 4. The 2 in [2] comes from linearization of the orthogonal subgraphs in the Coxeter diagram, as CDel node.pngCDel 2x.pngCDel node.png, with explicit branch order 2. The rhombic group, [2]+ (CDel node h2.pngCDel 2x.pngCDel node h2.png), half of the rectangular group, the point reflection symmetry, Z2, order 2.

Coxeter notation to allow a 1 place-holder for lower rank groups, so [1] is the same as [ ], and [1+] or [1]+ is the same as [ ]+ and Coxeter diagram CDel node h2.png.

The full p-gonal group [p], abstract dihedral group Dp, (nonabelian for p>2), of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter diagram CDel node.pngCDel p.pngCDel node.png. 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. For example, [[p]] adds a bisecting mirror to [p], and is isomorphic to [2p].

In the limit, going down to one dimensions, the full apeirogonal group is obtained when the angle goes to zero, so [∞], abstractly the infinite dihedral group D, represents two parallel mirrors and has a Coxeter diagram CDel node.pngCDel infin.pngCDel node.png. The apeirogonal group [∞]+, CDel node h2.pngCDel infin.pngCDel node h2.png, abstractly the infinite cyclic group Z, isomorphic to the additive group of the integers, is generated by a single nonzero translation.

In the hyperbolic plane, there is a full pseudogonal group [iπ/λ], and pseudogonal subgroup [iπ/λ]+, CDel node h2.pngCDel ultra.pngCDel node h2.png. These groups exist in regular infinite-sided polygons, with edge length λ. The mirrors are all orthogonal to a single line.

Group Intl Orbifold Coxeter Coxeter diagram Order Description
Finite
Zn n n• [n]+ CDel node h2.pngCDel n.pngCDel node h2.png n Cyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n.
Dn nm *n• [n] CDel node.pngCDel n.pngCDel node.png 2n Dihedral: cyclic with reflections. Abstract group Dihn, the dihedral group.
Affine
Z ∞• [∞]+ CDel node h2.pngCDel infin.pngCDel node h2.png Cyclic: apeirogonal group. Abstract group Z, the group of integers under addition.
Dih ∞m *∞• [∞] CDel node.pngCDel infin.pngCDel node.png Dihedral: parallel reflections. Abstract infinite dihedral group Dih.
Hyperbolic
Z [πi/λ]+ CDel node h2.pngCDel ultra.pngCDel node h2.png pseudogonal group
Dih [πi/λ] CDel node.pngCDel ultra.pngCDel node.png full pseudogonal group

Rank three groups[edit]

Finite family correspondence
Affine isomorphism and correspondences

In three dimensions, the full orthorhombic group [2,2], abtractly Z2×D2, order 8, represents three orthogonal mirrors, (also represented by Coxeter diagram as three separate dots CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png). It can also can be represented as a direct product [ ]×[ ]×[ ], but the [2,2] expression allows subgroups to be defined:

First there is a "semidirect" subgroup, the orthorhombic group, [2,2+] (CDel node.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h2.png), abstractly D1×Z2=Z2×Z2, of order 4. When the + superscript is given inside of the brackets, it means reflections generated only from the adjacent mirrors (as defined by the Coxeter diagram, CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png) are alternated. In general, the branch orders neighboring the + node must be even. In this case [2,2+] and [2+,2] represent two isomorphic subgroups that are geometrically distinct. The other subgroups are the pararhombic group [2,2]+ (CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png), also order 4, and finally the central group [2+,2+] (CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png) of order 2.

Next there is the full ortho-p-gonal group, [2,p] (CDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png), abstractly D1×Dp=Z2×Dp, of order 4p, representing two mirrors at a dihedral angle π/p, and both are orthogonal to a third mirror. It is also represented by Coxeter diagram as CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png.

The direct subgroup is called the para-p-gonal group, [2,p]+ (CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel p.pngCDel node h2.png), abstractly Dp, of order 2p, and another subgroup is [2,p+] (CDel node.pngCDel 2.pngCDel node h2.pngCDel p.pngCDel node h2.png) abstractly Z2×Zp, also of order 2p.

The full gyro-p-gonal group, [2+,2p] (CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel p.pngCDel node.png), abstractly D2p, of order 4p. The gyro-p-gonal group, [2+,2p+] (CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel p.pngCDel node h2.png), abstractly Z2p, of order 2p is a subgroup of both [2+,2p] and [2,2p+].

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. The Coxeter groups for these are called in Coxeter's bracket notation [3,3], [3,4], [3,5] called full tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, with orders of 24, 48, and 120. The front-to-back order can be reversed in the Coxeter notation, unlike the Schläfli symbol.

The tetrahedral group, [3,3], has a doubling [[3,3]] which maps the first and last mirrors onto each other, and this produces the [3,4] group.

In all these symmetries, alternate reflections can be removed producing the rotational tetrahedral, octahedral, and icosahedral groups of order 12, 24, and 60. The octahedral group also has a unique subgroup called the pyritohedral symmetry group, [3+,4], of order 12, with a mixture of rotational and reflectional symmetry.

In the Euclidean plane there's 3 fundamental reflective groups generated by 3 mirrors, represented by Coxeter diagrams CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png, CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png, and CDel node.pngCDel split1.pngCDel branch.png, and are given Coxeter notation as [4,4], [6,3], and [(3,3,3)]. The parentheses of the last group imply the diagram cycle, and also has a shorthand notation [3[3]].

[[4,4]] as a doubling of the [4,4] group produced the same symmetry rotated π/4 from the original set of mirrors.

Direct subgroups of rotational symmetry are: [4,4]+, [6,3]+, and [(3,3,3)]+. [4+,4] and [6,3+] are semidirect subgroups.

Subgroups[edit]

Given in Schönflies notation and Coxeter notation (orbifold notation), some low index point subgroups are:

Reflection Reflection
subgroups
Rotation subgroup Mixed Improper rotation Commutator
subgroup
C1v, [1]=[ ], CDel node.png, (*) C1, [1]+=[ ]+, CDel node h2.png, (11) S2, [2+,2+], CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png, (×) [ ]+
C2v, [2], CDel node.pngCDel 2.pngCDel node.png, (*22) [1+,2]=[1]=[ ], CDel node h0.pngCDel 2.pngCDel node.png (*) C2, [2]+, CDel node h2.pngCDel 2x.pngCDel node h2.png, (22) C2h, [2+,2], CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.png, (2*) S4, [4+,2+], CDel node h2.pngCDel 4.pngCDel node h4.pngCDel 2x.pngCDel node h2.png, (2×)
Cnv, [n], CDel node.pngCDel n.pngCDel node.png, (*nn) [1+,2n]=[n], CDel node h0.pngCDel 2x.pngCDel n.pngCDel node.png (*nn) Cn, [n]+, CDel node h2.pngCDel n.pngCDel node h2.png, (nn) Cnh, [n+,2], CDel node h2.pngCDel n.pngCDel node h2.pngCDel 2.pngCDel node.png, (n*) S2n, [2n+,2+], CDel node h2.pngCDel n.pngCDel node h4.pngCDel 2x.pngCDel node h2.png, (n×) [n]+, n odd
[n/2]+, n even
Dnh, [2,n], CDel node.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png, (*22n) [1+,2,n]=[1,n]=[n], CDel node h0.pngCDel 2x.pngCDel n.pngCDel node.png (*nn) Dn, [2,n]+, CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel n.pngCDel node h2.png, (22n) Dnd, [2+,2n], CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node.png, (2*n)
Td, [3,3], CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, (*332) T, [3,3]+, CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png, (332) [3,3]+, (332)
Oh, [4,3], CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, (*432) [1+,4,3]=[3,3], CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png (*332) O, [4,3]+, CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png, (432) Th, [3+,4], CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node.png, (3*2)
Ih, [5,3], CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png, (*532) I, [5,3]+, CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.png, (532) [5,3]+, (532)

Given in Coxeter notation (orbifold notation), some low index affine subgroups are:

Reflective
group
Reflective
subgroup
Mixed
subgroup
Rotation
subgroup
Improper rotation/
translation
Commutator
subgroup
[4,4], (*442) [1+,4,4], (*442)
[4,1+,4], (*2222)
[1+,4,4,1+], (*2222)
[4+,4], (4*2)
[(4,4,2+)], (2*22)
[1+,4,1+,4], (2*22)
[4,4]+, (442)
[1+,4,4+], (442)
[1+,4,1+4,1+], (2222)
[4+,4+], (22×) [4+,4+]+, (2222)
[6,3], (*632) [1+,6,3] = [3[3]], (*333) [3+,6], (3*3) [6,3]+, (632)
[1+,6,3+], (333)
[1+,6,3+], (333)

Rank four groups[edit]

Polychoral group tree.png
Subgroup relations

Point groups[edit]

Rank four groups defined the 4-dimensional point groups:

Subgroups[edit]

Space groups[edit]

Line groups[edit]

Rank four groups also defined the 3-dimensional line groups:

Duoprismatic group[edit]

Rank four groups defined the 4-dimensional duoprismatic groups. In the limit as p and q go to infinity, they degenerate into 2 dimensions and the wallpaper groups.

Wallpaper groups[edit]

Rank four groups also defined some of the 2-dimensional wallpaper groups, as limiting cases of the four-dimensional duoprism groups:

Subgroups of [∞,2,∞], (*2222) can be expressed down to its index 16 commutator subgroup:

Complex reflections[edit]

All subgroup relations on rank 2 Shephard groups.

Coxeter notation has 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. Complex reflection groups are called Shephard groups rather than Coxeter groups, and can be used to construct complex polytopes.

In , a rank 1 shephard group CDel pnode.png, order p, is represented as p[], []p or ]p[. It has a single generator, representing a 2π/p radian rotation in the Complex plane: .

Coxeter writes the rank 2 complex group, p[q]r represents Coxeter diagram CDel pnode.pngCDel 3.pngCDel q.pngCDel 3.pngCDel rnode.png. The p and r should only be suppressed if both are 2, which is the real case [q]. The order of a rank 2 group p[q]r is .[5]

The rank 2 solutions that generate complex polygons are: p[4]2 (p is 2,3,4,...), 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 with Coxeter diagrams CDel pnode.pngCDel 4.pngCDel node.png, CDel 3node.pngCDel 3.pngCDel 3node.png, CDel 3node.pngCDel 6.pngCDel node.png, CDel 3node.pngCDel 4.pngCDel 3node.png, CDel 4node.pngCDel 3.pngCDel 4node.png, CDel 3node.pngCDel 8.pngCDel node.png, CDel 4node.pngCDel 6.pngCDel node.png, CDel 4node.pngCDel 4.pngCDel 3node.png, CDel 3node.pngCDel 5.pngCDel 3node.png, CDel 5node.pngCDel 3.pngCDel 5node.png, CDel 3node.pngCDel 10.pngCDel node.png, CDel 5node.pngCDel 6.pngCDel node.png, CDel 5node.pngCDel 4.pngCDel 3node.png.

Some subgroup relations among infinite Shephard groups

Infinite groups are 3[12]2, 4[8]2, 6[6]2, 3[6]3, 6[4]3, 4[4]4, and 6[3]6 or CDel 3node.pngCDel 12.pngCDel node.png, CDel 4node.pngCDel 8.pngCDel node.png, CDel 6node.pngCDel 6.pngCDel node.png, CDel 3node.pngCDel 6.pngCDel 3node.png, CDel 6node.pngCDel 4.pngCDel 3node.png, CDel 4node.pngCDel 4.pngCDel 4node.png, CDel 6node.pngCDel 3.pngCDel 6node.png.

Index 2 subgroups exists by removing a real reflection: p[2q]2p[q]p. Also index r subgroups exist for 4 branches: p[4]rp[r]p.

For the infinite family p[4]2, for any p = 2, 3, 4,..., there are two subgroups: p[4]2 → [p], index p, while and p[4]2p[]×p[], index 2.

Notes[edit]

  1. ^ Conway, 2003, p.46, Table 4.2 Chiral groups II
  2. ^ Coxeter and Moser, 1980, Sec 9.5 Commutator subgroup, p. 124–126
  3. ^ Norman W. Johnson, Asia Ivić Weiss, Quaternionic modular groups, Linear Algebra and its Applications, Volume 295, Issues 1–3, 1 July 1999, Pages 159–189 [1]
  4. ^ The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF [2]
  5. ^ Coxeter, Regular Complex Polytopes, 9.7 Two-generator subgroups reflections. pp. 178–179

References[edit]