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3 Compatibility
 3.1 The compatibility condition (C)
 3.2 Compatible elements
 3.3 Compatible subgroups

3 Compatibility

3.1 The compatibility condition (C)

A subgroup F\le\mathrm{Aut}(B_{d,k}) satifies the compatibility condition (C) if and only if \mathrm{U}_{k}(F) is locally action isomorphic to F, see [Tor20, Proposition 3.8]. The term compatibility comes from the following translation of this condition into properties of the (k-1)-local actions of elements of F: The group F satisfies (C) if and only if

\forall \alpha\in F\ \forall\omega\in\Omega\ \exists\beta\in F:\ \sigma_{k-1}(\alpha,b)=\sigma_{k-1}(\beta,b_{\omega}),\ \sigma_{k-1}(\alpha,b_{\omega})=\sigma_{k-1}(\beta,b).

3.2 Compatible elements

This section is concerned with testing compatibility of two given elements (see AreCompatibleBallElements (3.2-1)) and finding an/all elements that is/are compatible with a given one (see CompatibleBallElement (3.2-2), CompatibilitySet (3.2-3)).

3.2-1 AreCompatibleBallElements
‣ AreCompatibleBallElements( d, k, aut1, aut2, dir )( function )

Returns: true if aut1 and aut2 are compatible with each other in direction dir, and false otherwise.

The arguments of this method are a degree d \in\mathbb{N}_{\ge 3}, a radius k \in\mathbb{N}, two automorphisms aut1, aut2 \in\mathrm{Aut}(B_{d,k}), and a direction dir \in[1..d].

gap> AreCompatibleBallElements(3,1,(1,2),(1,2,3),1);
true
gap> AreCompatibleBallElements(3,1,(1,2),(1,2,3),2);
false
gap> a:=(1,3,5)(2,4,6);; a in AutBall(3,2);
true
gap> LocalAction(1,3,2,a,[]); LocalAction(1,3,2,a,[1]);
(1,2,3)
(1,2)
gap> b:=(1,4)(2,3);; b in AutBall(3,2);
true
gap> LocalAction(1,3,2,b,[]); LocalAction(1,3,2,b,[1]);
(1,2)
(1,2,3)
gap> AreCompatibleBallElements(3,2,a,b,1);
true
gap> AreCompatibleBallElements(3,2,a,b,3);
false

3.2-2 CompatibleBallElement
‣ CompatibleBallElement( F, aut, dir )( function )

Returns: an element of F that is compatible with aut in direction dir if one exists, and fail otherwise.

The arguments of this method are a local action F \le\mathrm{Aut}(B_{d,k}), an element aut \in F, and a direction dir \in[1..d].

gap> mt:=RandomSource(IsMersenneTwister,1);;
gap> a:=Random(mt,AutBall(5,1)); dir:=Random(mt,[1..5]);
(1,2,5,4,3)
4
gap> CompatibleBallElement(AutBall(5,1),a,dir);
(1,2,5,4,3)
gap> a:=(1,3,5)(2,4,6);; a in AutBall(3,2);
true
gap> CompatibleBallElement(AutBall(3,2),a,1);
(1,4,2,3)

3.2-3 CompatibilitySet
‣ CompatibilitySet( F, aut, dir )( operation )
‣ CompatibilitySet( F, aut, dirs )( operation )
for the arguments F, aut, dir

Returns: the list of elements of F that are compatible with aut in direction dir.

The arguments of this method are a local action F of \le\mathrm{Aut}(B_{d,k}), an automorphism aut \in F, and a direction dir \in[1..d].

for the arguments F, aut, dirs

Returns: the list of elements of F that are compatible with aut in all directions of dirs.

The arguments of this method are a local action F of \le\mathrm{Aut}(B_{d,k}), an automorphism aut \in F, and a sublist of directions dirs \subseteq[1..d].

gap> F:=LocalAction(4,1,TransitiveGroup(4,3));
D(4)
gap> G:=LocalAction(4,1,SymmetricGroup(4));
Sym( [ 1 .. 4 ] )
gap> aut:=(1,3);; aut in F;
true
gap> CompatibilitySet(G,aut,1);
RightCoset(Sym( [ 2 .. 4 ] ),(1,3))
gap> CompatibilitySet(F,aut,1);
RightCoset(Group([ (2,4) ]),(1,3))
gap> CompatibilitySet(F,aut,[1,3]);
RightCoset(Group([ (2,4) ]),(1,3))
gap> CompatibilitySet(F,aut,[1,2]);
RightCoset(Group(()),(1,3))

3.2-4 AssembleAutomorphism
‣ AssembleAutomorphism( d, k, auts )( function )

Returns: the automorphism (aut,(auts[i])_{i=1}^{d}) of B_{d,k+1}, where aut is implicit in (auts[i])_{i=1}^{d}.

The arguments of this method are a degree d \in\mathbb{N}_{\ge 3}, a radius k \in\mathbb{N}, and a list auts of d automorphisms (auts[i])_{i=1}^{d} of B_{d,k} which comes from an element (aut,(auts[i])_{i=1}^{d}) of \mathrm{Aut}(B_{d,k+1}).

gap> mt:=RandomSource(IsMersenneTwister,1);;
gap> aut:=Random(mt,AutBall(3,2));
(1,4,5,2,3,6)
gap> auts:=[];;
gap> for i in [1..3] do auts[i]:=CompatibleBallElement(AutBall(3,2),aut,i); od;
gap> auts;
[ (1,4,6,2,3,5), (1,3,6,2,4,5), (1,5)(2,6) ]
gap> a:=AssembleAutomorphism(3,2,auts);
(1,7,9,3,5,11)(2,8,10,4,6,12)
gap> a in AutBall(3,3); 
true
gap> LocalAction(2,3,3,a,[]);
(1,4,5,2,3,6)

3.3 Compatible subgroups

Using the methods of Section 3.2, this section provides methods to test groups for the compatibility condition and search for compatible subgroups inside a given group, e.g. \mathrm{Aut}(B_{d,k}), or with a certain image under some projection.

3.3-1 MaximalCompatibleSubgroup
‣ MaximalCompatibleSubgroup( F )( attribute )

Returns: The local action C(F)\le\mathrm{Aut}(B_{d,k}), which is the maximal compatible subgroup of F.

The argument of this attribute is a local action F \le\mathrm{Aut}(B_{d,k}) (see IsLocalAction (2.1-1)).

gap> F:=LocalAction(3,1,Group((1,2)));
Group([ (1,2) ])
gap> MaximalCompatibleSubgroup(F);
Group([ (1,2) ])
gap> G:=LocalAction(3,2,Group((1,2)));
Group([ (1,2) ])
gap> MaximalCompatibleSubgroup(G);
Group(())

3.3-2 SatisfiesC
‣ SatisfiesC( F )( property )

Returns: true if F satisfies the compatibility condition (C), and false otherwise.

The argument of this property is a local action F \le\mathrm{Aut}(B_{d,k}) (see IsLocalAction (2.1-1)).

gap> D:=LocalActionDelta(3,SymmetricGroup(3));
Group([ (1,3,6)(2,4,5), (1,3)(2,4), (1,2)(3,4)(5,6) ])
gap> SatisfiesC(D);
true

3.3-3 CompatibleSubgroups
‣ CompatibleSubgroups( F )( function )

Returns: the list of all compatible subgroups of F.

The argument of this method is a local action F \le\mathrm{Aut}(B_{d,k}). This method calls AllSubgroups on F and is therefore slow. Use for instructional purposes on small examples only, and use ConjugacyClassRepsCompatibleSubgroups (3.3-4) or ConjugacyClassRepsCompatibleGroupsWithProjection (3.3-5) for computations.

gap> G:=LocalActionGamma(3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> list:=CompatibleSubgroups(G);
[ Group(()), Group([ (1,2)(3,5)(4,6) ]), Group([ (1,3)(2,4)(5,6) ]), 
  Group([ (1,6)(2,5)(3,4) ]), Group([ (1,4,5)(2,3,6) ]), 
  Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]) ]
gap> Size(list);
6
gap> Size(AllSubgroups(SymmetricGroup(3)));
6

3.3-4 ConjugacyClassRepsCompatibleSubgroups
‣ ConjugacyClassRepsCompatibleSubgroups( F )( attribute )

Returns: a list of compatible representatives of conjugacy classes of F that contain a compatible subgroup.

The argument of this method is a local action F of \mathrm{Aut}(B_{d,k}).

gap> ConjugacyClassRepsCompatibleSubgroups(AutBall(3,2));
[ Group(()), Group([ (1,2)(3,5)(4,6) ]), Group([ (1,4,5)(2,3,6) ]), 
  Group([ (3,5)(4,6), (1,2) ]), Group([ (1,2)(3,5)(4,6), (1,3,6)(2,4,5) ]), 
  Group([ (3,5)(4,6), (1,3,5)(2,4,6), (1,2)(3,4)(5,6) ]), 
  Group([ (1,2)(3,5)(4,6), (1,3,5)(2,4,6), (1,2)(5,6), (1,2)(3,4) ]), 
  Group([ (3,5)(4,6), (1,3,5)(2,4,6), (1,2)(5,6), (1,2)(3,4) ]), 
  Group([ (5,6), (3,4), (1,2), (1,3,5)(2,4,6), (3,5)(4,6) ]) ]

3.3-5 ConjugacyClassRepsCompatibleGroupsWithProjection
‣ ConjugacyClassRepsCompatibleGroupsWithProjection( l, F )( function )

Returns: a list of compatible representatives of conjugacy classes of \mathrm{Aut}(B_{d,l}) that contain a compatible group which projects to F \le\mathrm{Aut}(B_{d,r}).

The arguments of this method are a radius l \in\mathbb{N}, and a local action F \le\mathrm{Aut}(B_{d,k}) for some k\le l.

gap> S3:=LocalAction(3,1,SymmetricGroup(3));
Sym( [ 1 .. 3 ] )
gap> ConjugacyClassRepsCompatibleGroupsWithProjection(2,S3);
[ Group([ (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), 
  Group([ (1,2)(3,4)(5,6), (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), 
  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5,4,6) ]), 
  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5)(4,6) ]), 
  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (5,6), (3,5,4,6) ]) ]
gap> A3:=LocalAction(3,1,AlternatingGroup(3));
Alt( [ 1 .. 3 ] )
gap> ConjugacyClassRepsCompatibleGroupsWithProjection(2,A3);
[ Group([ (1,4,5)(2,3,6) ]) ]
gap> F:=SymmetricGroup(3);;
gap> rho:=SignHomomorphism(F);;
gap> H1:=LocalActionPi(2,3,F,rho,[0,1]);;
gap> H2:=LocalActionPi(2,3,F,rho,[1]);;
gap> Size(ConjugacyClassRepsCompatibleGroupsWithProjection(3,H1));
2
gap> Size(ConjugacyClassRepsCompatibleGroupsWithProjection(3,H2));
4
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