Quelle _Chapter_Signed_Permutations.xml Sprache: XML

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


<Chapter Label="Chapter_Signed_Permutations">
<Heading>Signed Permutations</Heading>

We provide <E>signed permutations</E>, that is permutations
that can additionally change the sign of their result.
<P/>
Assume <M>n \in \mathbb{N}</M>, then a signed permutation on <M>n</M> points
is a permutation <M>\pi</M> on <M>\{ 1 \ldots n \}</M> together with signs
<M>sgn : \{ 1 .. n \} \rightarrow \{-1,1\}</M>.
A signed permutation on <M>n</M> points acts on the set
<M>\{ -n \ldots 1, 1 \ldots n \}</M> by
<M> \omega ^ { (\pi, sgn) } = sgn(\omega)\cdot sgn(|\omega|^\pi) \cdot (|\omega|^\pi) </M>.
<P/>
We provide two representations of signed permutations, one as a list of images
<Ref Filt="IsSignedPermListRep" Label="for IsSignedPerm and IsPositionalObjectRep"/> and one formed as pair of a permutation and
a sign map <Ref Filt="IsSignedPermRep" Label="for IsSignedPerm and IsPositionalObjectRep"/>. Our benchmarks indicate that a list of
images is the better representation, and hence this is the default.
<P/>
To get started with signed permutations consider the following example
<Example><![CDATA[
gap> s := SignedPerm([2,-1]);
<signed permutation in list rep>
gap> 1 ^ s;
2
gap> 2 ^ s;
-1
gap> OnPoints(2, s);
-1
]]></Example>

<P/>
One can form groups out of signed permutations
<P/>
<Example><![CDATA[
gap> r := SignedPerm([-1,3,-2,4]);; t := SignedPerm([3,1,4,2]);;
gap> G := Group(r,t);
<group with 2 generators>
gap> Size(G);
32
gap> Orbit(G, 1, OnPoints);
[ 1, -1, 3, -3, -2, 4, 2, -4 ]
gap> Stabilizer(G, 1, OnPoints);
<group of size 4 with 9 generators>
]]></Example>

<P/>
Note that currently the package does not make an effort to exploit the special
structure of signed permutation groups as permutation groups.
<P/>
<Section Label="Chapter_Signed_Permutations_Section_Different_Representations">
<Heading>Different Representations</Heading>

<P/>
To create signed permutations in the different representations, we provide a constructor.
<Example><![CDATA[
gap> r := NewSignedPerm(IsSignedPermRep, [-1,3,-2,4]);;
gap> t := SignedPerm(IsSignedPermRep, [3,1,4,2]);;
gap> G := Group(r,t);
<group with 2 generators>
gap> Size(G);
32
gap> r := NewSignedPerm(IsSignedPermListRep, [-1,3,-2,4]);;
gap> t := SignedPerm(IsSignedPermListRep, [3,1,4,2]);;
gap> G := Group(r,t);
<group with 2 generators>
gap> Size(G);
32
]]></Example>

<P/>
</Section>

<Section Label="Chapter_Signed_Permutations_Section_Low-Level_Descriptions">
<Heading>Low-Level Descriptions</Heading>

<ManSection>
  <Filt Arg="arg" Name="IsSignedPerm" Label="for IsAssociativeElement andIsExtLElement andIsExtRElement andIsMultiplicativeElement andIsMultiplicativeElementWithOne andIsMultiplicativeElementWithInverse andIsFiniteOrderElement"/>
<Returns><K>true</K> or <K>false</K>
</Returns>
<Description>
Category of signed permutations
</Description>
</ManSection>

<ManSection>
  <Oper Arg="perm" Name="ListSignedPerm" Label="for IsSignedPerm"/>
<Description>
Convert a signed permutation into a list of images, equivalent
to List([1..LargestMovedPoint(s)], x -> x^s);
</Description>
</ManSection>

<ManSection>
  <Oper Arg="arg1,arg2" Name="ListSignedPerm" Label="for IsSignedPerm, IsPosInt"/>
<Description>
Convert a signed permutation to a list of images of length <A>len</A>.
Arguments perm, len
</Description>
</ManSection>

<ManSection>
  <Func Arg="arg" Name="SignedPerm" />
<Description>
Given a list of signed images create a signed permutation object
in <Ref Filt="IsSignedPermListRep" Label="for IsSignedPerm and IsPositionalObjectRep"/>.
</Description>
</ManSection>

<ManSection>
  <Constr Arg="arg1,arg2" Name="NewSignedPerm" Label="for IsSignedPerm, IsList"/>
<Description>
<P/>
</Description>
</ManSection>

<ManSection>
  <Constr Arg="arg1,arg2,arg3" Name="NewSignedPerm" Label="for IsSignedPerm, IsPerm, IsList"/>
<Description>
<P/>
</Description>
</ManSection>

<ManSection>
  <Filt Arg="arg" Name="IsSignedPermRep" Label="for IsSignedPerm and IsPositionalObjectRep"/>
<Returns><K>true</K> or <K>false</K>
</Returns>
<Description>
Representation of signed permutations as a permutation and a vector of signs.
</Description>
</ManSection>

<ManSection>
  <Filt Arg="arg" Name="IsSignedPermListRep" Label="for IsSignedPerm and IsPositionalObjectRep"/>
<Returns><K>true</K> or <K>false</K>
</Returns>
<Description>
Representation of signed permutations as a list of signed images
</Description>
</ManSection>

<ManSection>
  <Func Arg="arg" Name="OnPosPoints" />
<Description>
Only act as a permutation on <M>\{ 1\ldots n\}</M>
</Description>
</ManSection>

<ManSection>
  <Attr Arg="arg" Name="LargestMovedPoint" Label="for IsSignedPerm"/>
<Description>
The largest point that is moved by the signed permutation, where moving includes
changing the sign.
</Description>
</ManSection>

<ManSection>
  <Func Arg="arg" Name="RandomSignedPermList" />
<Description>
Create a random list of images that can be
used to create a signed permutation.
</Description>
</ManSection>

<ManSection>
  <Func Arg="arg" Name="RandomSignedPerm" />
<Description>
Create a random signed permutation
</Description>
</ManSection>

</Section>

</Chapter>

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