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<?xml version="1.0" encoding="UTF-8"?>


<Chapter Label="Chapter_Orbital_Structures">
<Heading>Orbital Structures</Heading>


The functions for orbital structures are based on recent work in permutation
group algorithms. An orbital structure contains information about orbits and
stabilisers of a group acting on a set for the purposes of quickly
determining representatives, canonising elements, and transversal elements
(directed) orbitals (orbits of ordered pairs of elements of the domain), and
undirected orbitals, i.e. orbits of sets of size two.

<Section Label="Chapter_Orbital_Structures_Section_Examples">
<Heading>Examples</Heading>

To create an orbital structure we need generators for a group, a set, and an
action
<Example><![CDATA[
gap> os := OrbitalStructure([
> (1,13,4,14,5)(2,10,12,9,8)(3,7,15,6,11)(16,17,18,20,19),
> (1,2,3)(4,6,5)(7,10,13)(8,12,14)(9,11,15)(16,18,21)(17,19,20) ],
> [1..21],
> OnPoints);;
gap> OrbitalRepresentative(os, [16,15]);
[ 16, 1 ]
gap> c := OrbitalCanonizingElement(os, [16, 15]);
(1,10,9,5,15)(2,7,6,8,4)(3,13,14,11,12)(17,20,18,19,21)
gap> OnTuples(c, [16,15]);
[ 16, 1 ]
gap> UnorderedOrbitalRepresentative(os, [16,2]);
[ 1, 16 ]
gap> c := UnorderedOrbitalCanonizingElement(os, [16,15]);
(1,15)(2,4)(3,12)(5,10)(7,8)(11,13)(17,21)(19,20)
gap> OnSets(c, Set([16,15]));
[ 1, 16 ]
gap> AllOrbitalRepresentatives(os)
[ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 16 ],
 [ 1, 18 ], [ 1, 20 ], [ 16, 1 ], [ 16, 2 ], [ 16, 3 ], [ 16, 16 ], [ 16, 17 ] ]
gap> AllUnorderedOrbitalRepresentatives(os)
[ [ 1, 1 ], [ 1, 2 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 16 ], [ 1, 18 ],
 [ 1, 20 ], [ 16, 16 ], [ 16, 17 ] ]
]]></Example>


<ManSection>
 <Filt Arg="arg" Name="IsOrbitalStructure" Label="for IsObject"/>
<Returns><K>true</K> or <K>false</K>
</Returns>
<Description>

</Description>
</ManSection>

<ManSection>
 <Func Arg="gens, domain, act" Name="OrbitalStructure" />
<Returns>An orbital structure
</Returns>
<Description>
Given generators, a set, and an action function create an orbital structure.
An orbital structure contains a list of orbits of the group generated by
<A>gens</A> on <A>domain</A>, a hashmap that maps any element of <A>domain</A>
to the index of its orbit in the list of orbits. We choose the smallest element of
each orbit as representative.
For each orbit, the orbital structure also contains the stabilizer of the chosen
orbit representative, together with all orbits of that stabilizer on <A>domain</A>
with chosen representatives.
</Description>
</ManSection>

<ManSection>
 <Func Arg="arg" Name="OS_OrbitRepresentative" />
<Description>

</Description>
</ManSection>

<ManSection>
 <Func Arg="arg" Name="OS_CanonisingElement" />
<Description>

</Description>
</ManSection>

<ManSection>
 <Func Arg="arg" Name="OS_CanonisingElementAndRepresentative" />
<Description>

</Description>
</ManSection>

<ManSection>
 <Func Arg="arg" Name="OS_StabilizerOf" />
<Description>

</Description>
</ManSection>

<ManSection>
 <Func Arg="os, pair" Name="OrbitalRepresentative" />
<Returns>pair
</Returns>
<Description>
Given an orbital structure <A>os</A> and a pair <A>pair</A>
of elements of the domain that <A>os</A> is defined on,
returns a canonical representative of <A>pair</A> in its orbit
of ordered pairs.
</Description>
</ManSection>

<ManSection>
 <Func Arg="os" Name="AllOrbitalRepresentatives" />
<Description>
Return the set of canonical representatives of orbits
of pairs under the action of the orbital structure.
</Description>
</ManSection>

<ManSection>
 <Func Arg="os, pair" Name="OrbitalCanonizingElement" />
<Returns>a group element
</Returns>
<Description>
Given an orbital structure <A>os</A> and the pair <A>pair</A>
returns an element <M>g</M> of the group that maps <A>pair</A> to
<C>OrbitalRepresentative(os, pair)</C>.
</Description>
</ManSection>

<ManSection>
 <Func Arg="arg" Name="OrbitalCanonizingElementInverse" />
<Description>

</Description>
</ManSection>

<ManSection>
 <Func Arg="os, pair" Name="OrbitalTransversalIterator" />
<Returns>an iterator
</Returns>
<Description>
Given an orbital structure <A>os</A> and a pair <A>pair</A>,
returns an iterator that produces an element <C>g</C> for every
element <C>e</C> in the orbit such that
<C>OnTuples(OrbitalRepresentative(os, pair), g) = e</C>.
</Description>
</ManSection>

<ManSection>
 <Func Arg="os, pair" Name="UnorderedOrbitalRepresentative" />
<Returns>pair
</Returns>
<Description>
Given an orbital structure <A>os</A> and a pair <A>pair</A>
of elements of the domain that <A>os</A> is defined on,
returns a canonical representative of <A>pair</A> in its orbit
of sets.
</Description>
</ManSection>

<ManSection>
 <Func Arg="os" Name="AllUnorderedOrbitalRepresentatives" />
<Description>
Return the set of canonical representatives of orbits
of sets of size two under the action of the orbital structure.
</Description>
</ManSection>

<ManSection>
 <Func Arg="os, pair" Name="UnorderedOrbitalTransversalIterator" />
<Returns>an iterator
</Returns>
<Description>
Given an orbital structure <A>os</A> and a pair <A>pair</A>,
returns an iterator that produces an element <C>g</C> for every
element <C>e</C> in the orbit such that
<C>OnSets(UnorderedOrbitalRepresentative(os, pair), g) = e</C>.
</Description>
</ManSection>

<ManSection>
 <Func Arg="os, pair" Name="UnorderedOrbitalCanonizingElement" />
<Returns>a group element
</Returns>
<Description>
Given an orbital structure <A>os</A> and the pair <A>pair</A>
returns an element <M>g</M> of the group that maps <A>pair</A> to
<C>UnorderedOrbitalRepresentative(os, pair)</C>.
</Description>
</ManSection>

<ManSection>
 <Func Arg="arg" Name="UnorderedOrbitalCanonizingElementInverse" />
<Description>

</Description>
</ManSection>

</Section>

</Chapter>

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