Quelle _Chapter_Compatibility.xml Sprache: XML

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


<Chapter Label="Chapter_Compatibility">
<Heading>Compatibility</Heading>


<Section Label="Section_condition_C">
<Heading>The compatibility condition (C)</Heading>

A subgroup <Math>F\le\mathrm{Aut}(B_{d,k})</Math> satifies the compatibility condition (C) if and only if <Math>\mathrm{U}_{k}(F)</Math> is locally action isomorphic to <Math>F</Math>, see <Cite Key="Tor20" Where="Proposition 3.8"/>. The term <E>compatibility</E> comes from the following translation of this condition into properties of the <Math>(k-1)</Math>-local actions of elements of <Math>F</Math>: The group <Math>F</Math> satisfies (C) if and only if <Display>\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).</Display>
</Section>

<Section Label="Section_compatible_elements">
<Heading>Compatible elements</Heading>

This section is concerned with testing compatibility of two given elements (see <Ref Func="AreCompatibleBallElements"/>) and finding an/all elements that is/are compatible with a given one (see <Ref Func="CompatibleBallElement"/>, <Ref Func="CompatibilitySet"/>).
<ManSection>
 <Func Arg="d,k,aut1,aut2,dir" Name="AreCompatibleBallElements" />
<Returns> <K>true</K> if <A>aut1</A> and <A>aut2</A> are compatible with each other in direction <A>dir</A>, and <K>false</K> otherwise.

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

</Description>
</ManSection>


<Example><![CDATA[
gap> AreCompatibleBallElements(3,1,(1,2),(1,2,3),1);
true
gap> AreCompatibleBallElements(3,1,(1,2),(1,2,3),2);
false
]]></Example>


<Example><![CDATA[
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
]]></Example>

<ManSection>
 <Func Arg="F,aut,dir" Name="CompatibleBallElement" />
<Returns> an element of <A>F</A> that is compatible with <A>aut</A> in direction <A>dir</A> if one exists, and <K>fail</K> otherwise.

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

</Description>
</ManSection>


<Example><![CDATA[
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)
]]></Example>


<Example><![CDATA[
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)
]]></Example>

<ManSection Label="CompatibilitySet">
<Heading>CompatibilitySet</Heading>
 <Oper Arg="F,aut,dir" Name="CompatibilitySet" Label="for F, aut, dir"/>
 <Oper Arg="F,aut,dirs" Name="CompatibilitySet" Label="for F, aut, dirs"/>
<Description>

<List>
for the arguments <A>F</A>, <A>aut</A>, <A>dir</A>
<Item>
 Returns: the list of elements of <A>F</A> that are compatible with <A>aut</A> in direction <A>dir</A>.

 The arguments of this method are a local action <A>F</A> of <Math>\le\mathrm{Aut}(B_{d,k})</Math>, an automorphism <A>aut</A> <Math>\in F</Math>, and a direction <A>dir</A> <Math>\in</Math><C>[1..d]</C>.
</Item>
for the arguments <A>F</A>, <A>aut</A>, <A>dirs</A>
<Item>
 Returns: the list of elements of <A>F</A> that are compatible with <A>aut</A> in all directions of <A>dirs</A>.

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

</Description>
</ManSection>


<Example><![CDATA[
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))
]]></Example>


<ManSection>
 <Func Arg="d,k,auts" Name="AssembleAutomorphism" />
<Returns> the automorphism <Math>(</Math><C>aut</C><Math>,(</Math><A>auts</A><Math>[</Math><C>i</C><Math>])_{i=1}^{d})</Math> of <Math>B_{d,k+1}</Math>, where <C>aut</C> is implicit in <Math>(</Math><A>auts</A><Math>[</Math><C>i</C><Math>])_{i=1}^{d}</Math>.

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

</Description>
</ManSection>


<Example><![CDATA[
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)
]]></Example>

</Section>

<Section Label="Chapter_Compatibility_Section_Compatible_subgroups">
<Heading>Compatible subgroups</Heading>

Using the methods of Section <Ref Sect="Section_compatible_elements"/>, this section provides methods to test groups for the compatibility condition and search for compatible subgroups inside a given group, e.g. <Math>\mathrm{Aut}(B_{d,k})</Math>, or with a certain image under some projection.
<ManSection>
 <Attr Arg="F" Name="MaximalCompatibleSubgroup" Label="for IsLocalAction"/>
<Returns>The local action <Math>C(</Math><A>F</A><Math>)\le\mathrm{Aut}(B_{d,k})</Math>, which is the maximal compatible subgroup of <A>F</A>.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>).

</Description>
</ManSection>


<Example><![CDATA[
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(())
]]></Example>

<ManSection>
 <Prop Arg="F" Name="SatisfiesC" Label="for IsLocalAction"/>
<Returns><K>true</K> if <A>F</A> satisfies the compatibility condition (C), and <K>false</K> otherwise.

</Returns>
<Description>
The argument of this property is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>).

</Description>
</ManSection>


<Example><![CDATA[
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
]]></Example>

<ManSection>
 <Func Arg="F" Name="CompatibleSubgroups" />
<Returns>the list of all compatible subgroups of <A>F</A>.

</Returns>
<Description>
The argument of this method is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math>. This method calls <C>AllSubgroups</C> on <Math>F</Math> and is therefore slow. Use for instructional purposes on small examples only, and use <Ref Attr="ConjugacyClassRepsCompatibleSubgroups" Label="for IsLocalAction"/> or <Ref Func="ConjugacyClassRepsCompatibleGroupsWithProjection"/> for computations.

</Description>
</ManSection>


<Example><![CDATA[
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
]]></Example>

<ManSection>
 <Attr Arg="F" Name="ConjugacyClassRepsCompatibleSubgroups" Label="for IsLocalAction"/>
<Returns>a list of compatible representatives of conjugacy classes of <A>F</A> that contain a compatible subgroup.

</Returns>
<Description>
The argument of this method is a local action <A>F</A> of <Math>\mathrm{Aut}(B_{d,k})</Math>.

</Description>
</ManSection>


<Example><![CDATA[
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) ]) ]
]]></Example>

<ManSection>
 <Func Arg="l,F" Name="ConjugacyClassRepsCompatibleGroupsWithProjection" />
<Returns> a list of compatible representatives of conjugacy classes of <Math>\mathrm{Aut}(B_{d,l})</Math> that contain a compatible group which projects to <A>F</A> <Math>\le\mathrm{Aut}(B_{d,r})</Math>.


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

</Description>
</ManSection>


<Example><![CDATA[
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) ]) ]
]]></Example>


<Example><![CDATA[
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
]]></Example>

</Section>

</Chapter>

quality98%

¤ Dauer der Verarbeitung: 0.14 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.