Symmetries

abstract type AbstractSymmetry end

An abstract type representing a symmetry.

Interface

Subtypes T of AbstractSymmetry should implement a method to make their objects symm callable on any object upon which the symmetry can act. For example, if T represents a symmetry of a 3D embedding of a graph:

• if x represents a 3D point, then symm(x) should be the image of that point.
• if x represents a 3D basis of space, then symm(x) should be the image of that basis.

When the symmetry is naturally defined on a discrete set of objects, their image should be accessible by indexing on the symmetry. With the same example as before:

• if x is a vertex, symm[x] is the image of x by the symmetry.
• if i is the integer identifier of vertex x, symm[i] is the identifier of symm[x].

IdentitySymmetry <: AbstractSymmetry

Identity symmetry, i.e. such that for s::IdentitySymmetry, ∀x, s[x] == x.

SimpleSymmetry{K,T,D} <: AbstractSymmetry

Symmetry operation defined by a map of type T and an optional dict of type D accepting keys of type K.

SimpleSymmetry(map::T, dict::D) where {T,D}

Create a SimpleSymmetry{keytype(D),T,D} object symm such that symm[x] is

• map[dict[x]] if x isa keytype(D).
• map[x] otherwise, if x isa Integer.

SimpleSymmetry(map)

Create a SimpleSymmetry object symm such that symm[x] == map[x] for all x.

AbstractSymmetryGroup{T<:AbstractSymmetry}

An abstract type representing the set of symmetries of a graph.

Interface

Any AbstractSymmetryGroup type must define methods for Base functions unique, iterate, length and one such that, for any s of type <: AbstractSymmetryGroup{T}:

• s(i) is a representative on the symmetry orbit of i such that all elements on the orbit share the same representative. The representative should be an integer.
• unique(s) is an iterator over such representatives.
• iterating over s yields the list of symmetry operations symm, each represented as an object of type T (where T <:AbstractSymmetry is the parameter to typeof(s)). The identity symmetry should not be part of these yielded symm, except for the specific IncludingIdentity subtype of AbstractSymmetryGroup.
• one(s) is the identity symmetry of type T.

NoSymmetryGroup <: AbstractSymmetryGroup{IdentitySymmetry}

The trivial AbstractSymmetryGroup devoid of any symmetry operation. NoSymmetryGroup(num) creates a NoSymmetryGroup over num unique representatives.

IncludingIdentity{S<:AbstractSymmetry,T<:AbstractSymmetryGroup{S}} <: AbstractSymmetryGroup{S}

Wrapper around an AbstractSymmetry that explicitly includes the identity operation.