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## WreathProductElements.gd
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##
## This file is part of the WPE package.
##
## This file's authors include Friedrich Rober.
##
## Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
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# Wreath Product Elements:
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## <#GAPDoc Label="Territory">
## <ManSection>
## <Attr Name="Territory" Arg="x"/>
## <Description>
## returns a list, namely the territory of <A>x</A>.
## The argument <A>x</A> must be a wreath product element
## (see <Ref Sect="Wreath Cycle"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "Territory", IsWreathProductElement );
## <#GAPDoc Label="IsWreathCycle">
## <ManSection>
## <Attr Name="IsWreathCycle" Arg="x"/>
## <Description>
## returns true or false. Tests whether <A>x</A> is a wreath cycle.
## The argument <A>x</A> must be a wreath product element
## (see <Ref Sect="Wreath Cycle"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsWreathCycle", IsWreathProductElement );
## <#GAPDoc Label="IsSparseWreathCycle">
## <ManSection>
## <Attr Name="IsSparseWreathCycle" Arg="x"/>
## <Description>
## returns true or false. Tests whether <A>x</A> is a sparse wreath cycle.
## The argument <A>x</A> must be a wreath product element
## (see <Ref Sect="Sparse Wreath Cycle"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareProperty( "IsSparseWreathCycle", IsWreathProductElement );
## <#GAPDoc Label="WreathCycleDecomposition">
## <ManSection>
## <Attr Name="WreathCycleDecomposition" Arg="x"/>
## <Description>
## returns a list containing wreath cycles, namely the wreath cycle decomposition of <A>x</A>.
## The argument <A>x</A> must be a wreath product element
## (see <Ref Sect="Wreath Cycle"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "WreathCycleDecomposition", IsWreathProductElement );
# TODO: Replace these attributes
DeclareAttribute( "SparseWreathCycleConjugate", IsWreathProductElement );
DeclareAttribute( "ConjugatorWreathCycleToSparse", IsWreathProductElement );
## <#GAPDoc Label="Yade">
## <ManSection>
## <Attr Name="Yade" Arg="x, [i]"/>
## <Description>
## returns a group element, namely the yade of the wreath cycle <A>x</A>
## evaluated at the smallest territory point.
## If the optional argument <A>i</A> is provided,
## the function returns the yade evaluated at the point <A>i</A>.
## The argument <A>x</A> must be a wreath cycle
## and the optional argument <A>i</A> must be an integer
## from the territory of <A>x</A>
## (see <Ref Sect="Sparse Wreath Cycle"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareAttribute( "Yade", IsWreathCycle );
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# Display:
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## <#GAPDoc Label="DisplayOptionsForWreathProductElements">
## <ManSection>
## <Func Name="DisplayOptionsForWreathProductElements" Arg=""/>
## <Description>
## prints the current global display options for wreath product elements.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "DisplayOptionsForWreathProductElements" );
## <#GAPDoc Label="SetDisplayOptionsForWreathProductElements">
## <ManSection>
## <Func Name="SetDisplayOptionsForWreathProductElements" Arg="optrec"/>
## <Description>
## sets the current global display options for wreath product elements. <P/>
## The argument <A>optrec</A> must be a record with components that are valid display options. (see <
Ref Label="Display Functions"/>)
## The components for the current global display options are set to the values specified by the components in <A>optrec</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "SetDisplayOptionsForWreathProductElements" );
## <#GAPDoc Label="ResetDisplayOptionsForWreathProductElements">
## <ManSection>
## <Func Name="ResetDisplayOptionsForWreathProductElements" Arg=""/>
## <Description>
## resets the current global display options for wreath product elements to default.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "ResetDisplayOptionsForWreathProductElements" );
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# Components:
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## <#GAPDoc Label="ComponentsOfWreathProduct">
## <ManSection>
## <Func Name="ComponentsOfWreathProduct" Arg="W"/>
## <Description>
## returns a list of two groups [K, H], where <A>W = K wr H</A>.
## The argument <A>W</A> must be a wreath product
## (see <Ref Sect="Wreath Product"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "ComponentsOfWreathProduct" );
## <#GAPDoc Label="TopComponentOfWreathProductElement">
## <ManSection>
## <Func Name="TopComponentOfWreathProductElement" Arg="x"/>
## <Description>
## returns a group element, namely the top component of <A>x</A>.
## The argument <A>x</A> must be a wreath product element
## (see <Ref Sect="Wreath Product"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "TopComponentOfWreathProductElement" );
DeclareOperation( "WPE_TopComponent", [IsWreathProductElement] );
## <#GAPDoc Label="TopGroupOfWreathProduct">
## <ManSection>
## <Func Name="TopGroupOfWreathProduct" Arg="W"/>
## <Description>
## returns a group, namely the top group <M>\langle 1_K \rangle^m \times H</M> of the wreath product <M>W = K \wr H</M>
## (see <Ref Sect="Wreath Product"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "TopGroupOfWreathProduct" );
DeclareOperation( "WPE_TopGroup", [HasWreathProductInfo] );
DeclareOperation( "WPE_TopDegree", [IsWreathProductElement] );
## <#GAPDoc Label="BaseComponentOfWreathProductElement">
## <ManSection>
## <Func Name="BaseComponentOfWreathProductElement" Arg="x, [i]"/>
## <Description>
## returns a group element, namely the base component of <A>x</A>.
## If the optional argument <A>i</A> is provided,
## the function returns the <A>i</A>-th base component of <A>x</A>.
## The argument <A>x</A> must be a wreath product element
## and the optional argument <A>i</A> must be an integer
## (see <Ref Sect="Wreath Product"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "BaseComponentOfWreathProductElement" );
DeclareOperation( "WPE_BaseComponent", [IsWreathProductElement] );
## <#GAPDoc Label="BaseGroupOfWreathProduct">
## <ManSection>
## <Func Name="BaseGroupOfWreathProduct" Arg="W, [i]"/>
## <Description>
## returns a group, namely the base group <M>K^m \times \langle 1_H</M> of the wreath product <M>W = K \wr H</M>.
## If the optional argument <A>i</A> is provided,
## the function returns the <A>i</A>-th factor of the base group of <A>W</A>
## (see <Ref Sect="Wreath Product"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "BaseGroupOfWreathProduct" );
DeclareOperation( "WPE_BaseGroup", [HasWreathProductInfo] );
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# Isomorphism:
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## <#GAPDoc Label="IsomorphismWreathProduct">
## <ManSection>
## <Oper Name="IsomorphismWreathProduct" Arg="G"/>
## <Description>
## returns an isomorphism from a specialized wreath product <A>G</A>
## to a generic wreath product. <P/>
## <Example><![CDATA[
## gap> K := AlternatingGroup(5);;
## gap> H := SymmetricGroup(4);;
## gap> G := WreathProduct(K, H);
## <permutation group of size 311040000 with 10 generators>
## gap> iso := IsomorphismWreathProduct(G);;
## gap> W := Image(iso);
## <group of size 311040000 with 4 generators>
## ]]></Example>
## For an overview on wreath product representations in &GAP; see <Ref Sect="Wreath Product Representations"/>. <P/>
## In the background, it uses the low-level functions
## <C>ListWreathProductElement</C> and <C>WreathProductElementList</C>
## and wraps the <C>IsList</C> representations into <C>IsWreathProductElement</C> representations. <P/>
## For performant code, we recommend to use these low-level functions instead of <C>IsomorphismWreathProduct</C>.
## All functions for <C>IsWreathProductElement</C> also work on <C>IsList</C> objects that represent a wreath product element.
## However, it is not checked that the <C>IsList</C> object actually represents a wreath product element.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareOperation( "IsomorphismWreathProduct", [HasWreathProductInfo] );
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# Cycle Index:
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## <#GAPDoc Label="CycleIndexWreathProductProductAction">
## <ManSection>
## <Func Name="CycleIndexWreathProductProductAction" Arg="K, H"/>
## <Description>
## For two permutation groups <A>K</A> and <A>H</A>
## this function constructs the cycle index polynomial of
## the wreath product <M>K \wr H</M> in product action. <P/>
## The implementation is based on <Cite Key="HarrisonHigh"/> and <Cite Key="PalmerRobinson"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "CycleIndexWreathProductProductAction");
## <#GAPDoc Label="CycleIndexWreathProductImprimitiveAction">
## <ManSection>
## <Func Name="CycleIndexWreathProductImprimitiveAction" Arg="K, H"/>
## <Description>
## For two permutation groups <A>K</A> and <A>H</A>
## this function constructs the cycle index polynomial of
## the wreath product <M>K \wr H</M> in imprimitive action. <P/>
## The implementation is based on <Cite Key="Polya"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "CycleIndexWreathProductImprimitiveAction");
