<!-- 

  stab_orbs.xml    FinInG package documentation
                                                                   John Bamberg
                                                                   Anton Betten
                                                                  Philippe Cara
                                                                   Jan De Beule
                                                                 Michel Lavrauw
                                                                Max Neunhoeffer

  Copyright (C) 2018,	Colorado State University
						Università degli Studi di Padova
						Universiteit Gent
						University of St. Andrews
						University of Western Australia
                        Vrije Universiteit Brussel

This is the chapter of the documentation describing the stabilisers and orbits.

-->

<Chapter Label="acts_stab_orbs">
<Heading>Orbits, stabilisers and actions</Heading>

<Section>
<Heading>Orbits</Heading>
GAP provides generic functionality to compute orbits. These functions are, generally spoken, applicable to the
groups implemented in <Package>FinInG</Package>, combined with the appropriate action functions. However, the generic
functions applied in such situations are rather time consuming. <Package>FinInG</Package> therefore provides alternative
functions to compute orbits.

<ManSection>
<Oper Name="FiningOrbit" Arg="g, obj, act"/>
<Returns>The orbit of the object <A>obj</A> under the action <A>act</A> of the
group <A>g</A>.</Returns>
<Description>
The argument <A>obj</A> is either a subspace of a projective space, then combined with
the action function <F>OnProjSubspaces</F>, or a set of elements of a projective space, then
combined with the action function <F>OnSetsProjSubspaces</F>. The group <A>g</A> is a subgroup
of a collineation group of a projective space. In both cases the action function computes the action
of <A>el</A> under the group element <A>g</A>.
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_finingorbit1.include"> 
</Example>
The second example shows the possible use of <F>FiningOrbit</F> in combination with the action function
<F>OnSetsProjSubspaces</F>. Please note that this variant is probably not the most efficient way to compute all elliptic
quadrics contained in the parabolic quadric <M>ps</M>. Experiments show that for <M>q=5</M> the second variant takes an 
unreasonable amount of time. Also note that the second argument <A>el</A> must be
a set (and therefore it might be necessary to apply <F>Set</F> on a collection of elements).
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_finingorbit2.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Oper Name="FiningOrbits" Arg="g, set, act"/>
<Oper Name="FiningOrbits" Arg="g, coll"/>
<Returns>The orbits of the group <A>g</A> on <A>set</A> under the action of <A>act</A>.</Returns>
<Description>
The set is a set of elements of a projective space, the group <A>g</A> is a subgroup of the collineation group
of a projective space, and <A>act</A> is the function <F>OnProjSubspaces</F>. If <A>coll</A> is a collection of
elements of a projective space (i.e. not a list or set, but and object representing the collection of elements
of a given type, such as <F>Lines(PG(3,4))</F>), then the second versions returns the orbits of <A>g</A> on the elements
of <A>coll</A> under the action <F>OnProjSubspaces</F>.
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_finingorbits1.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Oper Name="FiningOrbitsDomain" Arg="g, coll, act"/>
<Returns>The orbits of the group <A>g</A> on a collection <A>coll</A> under the action of <A>act</A>.</Returns>
<Description>
The argument <A>coll</A> must be an object in the category <C>IsElementsOfIncidenceGeometry</C>, the argument <A>g</A>
can be any group, acting on the elements of <A>coll</A> through a suitable action function <A>act</A>. This operation
is inspired by the GAP operation <F>OrbitsDomain</F>. It computes the orbits of the group <A>g</A> on the set of 
elements in <A>coll</A> <E>assuming</E> that the set of elements of <A>coll</A> is closed under that action of <A>g</A>.
There is no check whether the assumption is correct, so this operation should typically be used when it is known that
the assumption is correct. The operation is generic in the sense that it can be used for different types of 
incidence geometries as long as the triple arguments consist of a suitable action of <A>g</A> on the elements of <A>coll</A>.
In general, if the assumption is correct, this operation will be faster than <F>FiningOrbits</F>.
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_finingorbitsdomain.include"> 
</Example>
</Description>
</ManSection>

</Section>

<Section>
<Heading>Stabilisers</Heading>

The GAP function <F>Stabilizer</F> is a generic function to compute stabilisers of one object (or sets or tuples etc. of objects)
under a group, using a specified action function. This generic function can be used together with the in <Package>FinInG</Package>
implemented groups and elements of geometries. However, computing time can be very long, already in small geometries.

<Example>
<#Include SYSTEM "../examples/include/stab_orbs_example1.include"> 
</Example>

The packages <Package>GenSS</Package> and <Package>orb</Package> required by <Package>FinInG</Package> provide 
efficient operations to compute stabilisers, and <Package>FinInG</Package> provides functionality to use 
these operations for the particular groups and (elements) of geometries.

<ManSection>
<Oper Name="FiningStabiliser" Arg="g, el"/>
<Returns>The subgroup of <A>g</A> stabilising the element <A>el</A></Returns>
<Description>The argument <A>g</A> is a group of collineations acting on the element <A>el</A>, being a
subspace of a projective space (and hence, all elements of a Lie geometry are allowed as second argument).
This operation relies on the <Package>GenSS</Package> operation <F>Stab</F>.
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_finingstabiliser.include"> 
</Example>
</Description>
</ManSection>


<ManSection>
<Oper Name="FiningStabiliserOrb" Arg="g, el"/>
<Returns>The subgroup of <A>g</A> stabilising the element <A>el</A></Returns>
<Description>The argument <A>g</A> is a group of collineations acting on the element <A>el</A>, being a
subspace of a projective space (and hence, all elements of a Lie geometry are allowed as second argument).
This operation relies on some particular <Package>orb</Package> functionality.
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_finingstabiliserorb.include"> 
</Example>
</Description>
</ManSection>

A small example shows the difference in computing time. Clearly the <F>FiningStabiliserOrb</F> is the fastest 
way to compute stabilizers of one element.
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_timing1.include"> 
</Example>


<ManSection>
<Oper Name="FiningSetwiseStabiliser" Arg="g, els"/>
<Returns>The subgroup of <A>g</A> stabilising the set <A>els</A></Returns>
<Description>The argument <A>g</A> is a group of collineations acting on the element <A>el</A>, being a
subspace of a projective space (and hence, all elements of a Lie geometry are allowed as second argument).
The argument <A>els</A> is a set of elements of the same type of the same Lie geometry, the elements are all in 
the category <C>IsSubspaceOfProjectiveSpace</C>. The underlying action function is assumed to be
<F>OnProjSubspaces</F>
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_finingsetwisestabiliser.include"> 
</Example>
</Description>
</ManSection>
Computing the setwise stabiliser under a group is also possible using <F>Stabilizer</F>. But, not surprisingly, the computing time can be very long.
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_setwisegeneric.include"> 
</Example>
The package <Package>GenSS</Package> provides an efficient operations to compute setwise stabilisers. This is why <Package>FinInG</Package> provides functionality, such as <F>FiningSetwiseStabiliser</F>, to use these <Package>GenSS</Package> operations for the particular groups and (elements) of geometries.
A small example shows the difference in computing time. 
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_timing2.include"> 
</Example>

For subspaces of projective spaces we can be even more efficient by writing down generators of the stabiliser and then return the generated subgroup.

<ManSection>
<Oper Name="StabiliserGroupOfSubspace" Arg="sub"/>
<Returns>The subgroup of the collineation group of <A>AmbientSpace(sub)</A> stabilising the subspace <A>sub</A></Returns>
<Description>The argument <A>sub</A> is a subspace of a projective space.

</Description>
</ManSection>
<ManSection>
<Oper Name="ProjectiveStabiliserGroupOfSubspace" Arg="sub"/>
<Returns>The subgroup of the projectivity group of <A>AmbientSpace(sub)</A> stabilising the subspace <A>sub</A></Returns>
<Description>The argument <A>sub</A> is a subspace of a projective space.

</Description>
</ManSection>
<ManSection>
<Oper Name="SpecialProjectiveStabiliserGroupOfSubspace" Arg="sub"/>
<Returns>The subgroup of the special projectivity group of <A>AmbientSpace(sub)</A> stabilising the subspace <A>sub</A></Returns>
<Description>The argument <A>sub</A> is a subspace of a projective space.

<Example>
<#Include SYSTEM "../examples/include/stab_orbs_stabofsubspace.include"> 
</Example>
</Description>
</ManSection>


</Section>

<Section>
<Heading>Actions and nice monomorphisms revisited</Heading>

GAP provides generic functions to compute action homomorphisms and their images for arbitrary groups. These
functions are applicable on the projective groups implemented in <Package>FinInG</Package>.

<ManSection>
<Heading>Action functions</Heading>
<Func Name="OnProjSubspaces" Arg="el, g"/>
<Func Name="OnProjSubspacesExtended" Arg="el, g"/>
<Func Name="OnSetsProjSubspaces" Arg="set, g"/>
<Returns>a element of a Lie geometry</Returns>
<Description>
Let <A>el</A> be an element of any Lie geometry, and <A>g</A> an element of a projective group
acting on the elements of the ambient Lie geometry of <A>el</A>. Then then <F>OnProjSubspaces</F>
will return simply the image of <A>el</A> under <A>g</A>. When <A>g</A> is an element of the correlation/collineation group
of a projective space, <F>OnProjSubspacesExtended</F> returns the image of <A>el</A> under <A>g</A>. Finally,
when <A>set</A> is a set of elements of a Lie geometry, <F>OnSetsProjSubspaces</F> returns the set of images
under <A>g</A>. <F>OnProjSubspaces</F> is also explained in <Ref Sect="proj:onprojsubspaces"/>,
<F>OnProjSubspacesExtended</F> is also explained in <Ref Sect="proj:onprojsubspacesextended"/>.
</Description>
</ManSection>

<ManSection>
<Heading>Generic GAP functions</Heading>
<Oper Name="ActionHomomorphism" Arg="g, S, act"/>
<Oper Name="Action" Arg="g, S, act"/>
<Description><A>g</A> is a projective group, <A>S</A> is a set or a collection of elements, <A>act</A>
is an action function. <F>Action</F> simply returns <F>Image(hom)</F>, if <A>hom</A> is the result
of <F>ActionHomomorphism</F>. The examples are self-explanatory.
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_action1.include">
</Example>
</Description>
</ManSection>

A nice monomorphism of a group <M>G</M> is roughly just a permutation representation of <M>G</M> on a
suitable action domain. An easy example is the permutation action of the full collineation
group of a projective space on its points.<P/>

<ManSection>
<Attr Name="NiceMonomorphism" Arg="group"/>
<Returns>A group homomorphism</Returns>
<Description>
This is a generic GAP function, and returns a homomorphism to a "better" representation.
</Description>
</ManSection>

<ManSection>
<Attr Name="NiceObject" Arg="group"/>
<Returns>A permutation group</Returns>
<Description>
<A>group</A> is a projective group. The object this operation returns is actually equivalent with <F>Image(NiceMonomorphism(group))</F>.
</Description>
</ManSection>

<Subsection>
<Heading>Different behaviour for different collineation groups</Heading>
For the different Lie geometries implemented in <Package>FinInG</Package>, nicemonomorphisms
are (necessarily) treated in a different way. As the aim of a nicemonomorphism of
group <M>G</M> is to provide a permutation representation, such that efficient algorithms for
permutation groups become available for certain operations applicable on <M>G</M>, clearly the
efficiency will be increased if the degree of the permutation representation is as small as
possible.<P/>
For the collineation group, projectivity group and special projectivity group of a projective
space, it is clear that the smallest degree permutation representation is the action of the group
on the projective points. In principle, one could also consider the action on the hyperplanes.
For the collineation group, similarity group and isometry group of a classical polar space,
in most cases, the smallest degree permutation representation is the action on the points. A notorious
exception to this is the hermitian polar space in three dimensions, of which the number of lines is smaller than
the number of points, and hence of which the smallest degree permutation representation is the action
of the group on the lines. When constructing a collineation group (or (special) projectivity group) of
a projective space, the nicemonomorphism is not computed. It is only computed when needed. The reason is that from the underlying field and dimension, the underlying
projective space can be determined at any time, and hence the smallest degree representation can be computed.
For the collineation groups (and similarity and isometry groups) of classical polar spaces, this behaviour is
different. Indeed, given a group of collineations, from the underlying field and dimension, the original
polar space can not be determined. Of course one could consider the action on the points of the underlying
projective space, but typically the number of points of a classical polar space is much smaller than the number
points of the underlying projective space. This explains why, currently, a nice monomorphism is computed
at the moment a collineation group of a classical polar space is computed. As a consequence, just asking the
collineation group of a polar space can be time consuming.
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_behaviour1.include">
</Example>
</Subsection>

<ManSection>
<Oper Name="SetParent" Arg="group"/>
<Description>
Assume that <M>G</M> is a group of collineations. As mentioned already, from the underlying field
and dimension, only the underlying projective space can be determined. An operation like <F>Order</F>
requires a nice monomorphism, so for an arbitrary group <M>G</M>, the action on the points of the
underlying projective space will be computed, which can be time consuming for large projective spaces.
However, if it is known that <M>G</M> is a subgroup of another collineation group <M>H</M>, this group
<M>H</M> can be set as a parent group for <M>G</M>. If a nice monomorphism is available for <M>H</M>, it
will become available for <M>G</M>. In the example we construct the collineation group of the hermitian
polar space <M>H(3,81)</M>. As explained, a nice monomorphism is computed upon construction. Then
construct a group generated by two random elements of this collineation group of <M>H(3,81)</M>, and compute
its order. Without further information, it will be assumed by the system that this new group is a subgroup
of the collineation group of <M>PG(3,81)</M>, and a nice monomorphism will be computed through this group.
In the second part we set the parent group as the collineation group of <M>H(3,81)</M>, and compute the order
again. Compare the different timings.
<Example>
<#Include SYSTEM "../examples/include/stab_orbs_behaviour2.include">
</Example>
</Description>
</ManSection>

</Section>

</Chapter>
