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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the declaration of those functions that are used
## to construct maps (mostly fusion maps and power maps).
##
## 1. Maps Concerning Character Tables
## 2. Power Maps
## 3. Class Fusions between Character Tables
## 4. Utilities for Parametrized Maps
## 5. Subroutines for the Construction of Power Maps
## 6. Subroutines for the Construction of Class Fusions
##
#############################################################################
##
## 1. Maps Concerning Character Tables
##
## <#GAPDoc Label="[1]{ctblmaps}">
## Besides the characters, <E>power maps</E> are an important part of a
## character table, see Section <Ref Sect="Power Maps"/>.
## Often their computation is not easy, and if the table has no access to
## the underlying group then in general they cannot be obtained from the
## matrix of irreducible characters;
## so it is useful to store them on the table.
## <P/>
## If not only a single table is considered but different tables of a group
## and a subgroup or of a group and a factor group are used,
## also <E>class fusion maps</E>
## (see Section <Ref Sect="Class Fusions between Character Tables"/>)
## must be known to get information about the embedding or simply to induce
## or restrict characters,
## see Section <Ref Sect="Restricted and Induced Class Functions"/>).
## <P/>
## These are examples of functions from conjugacy classes which will be
## called <E>maps</E> in the following.
## (This should not be confused with the term mapping,
## cf. Chapter <Ref Chap="Mappings"/>.)
## In &GAP;, maps are represented by lists.
## Also each character, each list of element orders, of centralizer orders,
## or of class lengths are maps,
## and the list returned by <Ref Func="ListPerm"/>,
## when this function is called with a permutation of classes, is a map.
## <P/>
## When maps are constructed without access to a group, often one only knows
## that the image of a given class is contained in a set of possible images,
## e. g., that the image of a class under a subgroup fusion is in the set of
## all classes with the same element order.
## Using further information, such as centralizer orders, power maps and the
## restriction of characters, the sets of possible images can be restricted
## further.
## In many cases, at the end the images are uniquely determined.
## <P/>
## Because of this approach, many functions in this chapter work not only
## with maps but with <E>parametrized maps</E>
## (or <E>paramaps</E> for short).
## More about parametrized maps can be found
## in Section <Ref Sect="Parametrized Maps"/>.
## <P/>
## The implementation follows <Cite Key="Bre91"/>,
## a description of the main ideas together with several examples
## can be found in <Cite Key="Bre99"/>.
## <#/GAPDoc>
##
#############################################################################
##
## 2. Power Maps
##
## <#GAPDoc Label="[2]{ctblmaps}">
## The <M>n</M>-th power map of a character table is represented by a list
## that stores at position <M>i</M> the position of the class containing
## the <M>n</M>-th powers of the elements in the <M>i</M>-th class.
## The <M>n</M>-th power map can be composed from the power maps of the
## prime divisors of <M>n</M>,
## so usually only power maps for primes are actually stored in the
## character table.
## <P/>
## For an ordinary character table <A>tbl</A> with access to its underlying
## group <M>G</M>,
## the <M>p</M>-th power map of <A>tbl</A> can be computed using the
## identification of the conjugacy classes of <M>G</M> with the classes of
## <A>tbl</A>.
## For an ordinary character table without access to a group,
## in general the <M>p</M>-th power maps (and hence also the element orders)
## for prime divisors <M>p</M> of the group order are not uniquely
## determined by the matrix of irreducible characters.
## So only necessary conditions can be checked in this case,
## which in general yields only a list of several possibilities for the
## desired power map.
## Character tables of the &GAP; character table library store all
## <M>p</M>-th power maps for prime divisors <M>p</M> of the group order.
## <P/>
## Power maps of Brauer tables can be derived from the power maps of the
## underlying ordinary tables.
## <P/>
## For (computing and) accessing the <M>n</M>-th power map of a character
## table, <Ref Oper="PowerMap"/> can be used;
## if the <M>n</M>-th power map cannot be uniquely determined then
## <Ref Oper="PowerMap"/> returns <K>fail</K>.
## <P/>
## The list of all possible <M>p</M>-th power maps of a table in the sense
## that certain necessary conditions are satisfied can be computed with
## <Ref Oper="PossiblePowerMaps"/>.
## This provides a default strategy, the subroutines are listed in
## Section <Ref Sect="Subroutines for the Construction of Power Maps"/>.
## <#/GAPDoc>
##
#############################################################################
##
#O PowerMap( <tbl>, <n>[, <class>] )
#O PowerMapOp( <tbl>, <n>[, <class>] )
#A ComputedPowerMaps( <tbl> )
##
## <#GAPDoc Label="PowerMap">
## <ManSection>
## <Oper Name="PowerMap" Arg='tbl, n[, class]'/>
## <Oper Name="PowerMapOp" Arg='tbl, n[, class]'/>
## <Attr Name="ComputedPowerMaps" Arg='tbl'/>
##
## <Description>
## Called with first argument a character table <A>tbl</A>
## and second argument an integer <A>n</A>,
## <Ref Oper="PowerMap"/> returns the <A>n</A>-th power map of <A>tbl</A>.
## This is a list containing at position <M>i</M> the position of the class
## of <A>n</A>-th powers of the elements in the <M>i</M>-th class of
## <A>tbl</A>.
## <P/>
## If the additional third argument <A>class</A> is present then the
## position of <A>n</A>-th powers of the <A>class</A>-th class is returned.
## <P/>
## If the <A>n</A>-th power map is not uniquely determined by <A>tbl</A>
## then <K>fail</K> is returned.
## This can happen only if <A>tbl</A> has no access to its underlying group.
## <P/>
## The power maps of <A>tbl</A> that were computed already by
## <Ref Oper="PowerMap"/> are stored in <A>tbl</A> as value of the attribute
## <Ref Attr="ComputedPowerMaps"/>,
## the <M>n</M>-th power map at position <M>n</M>.
## <Ref Oper="PowerMap"/> checks whether the desired power map is already
## stored, computes it using the operation <Ref Oper="PowerMapOp"/> if it is
## not yet known, and stores it.
## So methods for the computation of power maps can be installed for
## the operation <Ref Oper="PowerMapOp"/>.
## <!-- % For power maps of groups, see <Ref Attr="PowerMapOfGroup"/>. -->
## <P/>
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "L3(2)" );;
## gap> ComputedPowerMaps( tbl );
## [ , [ 1, 1, 3, 2, 5, 6 ], [ 1, 2, 1, 4, 6, 5 ],,,,
## [ 1, 2, 3, 4, 1, 1 ] ]
## gap> PowerMap( tbl, 5 );
## [ 1, 2, 3, 4, 6, 5 ]
## gap> ComputedPowerMaps( tbl );
## [ , [ 1, 1, 3, 2, 5, 6 ], [ 1, 2, 1, 4, 6, 5 ],, [ 1, 2, 3, 4, 6, 5 ],
## , [ 1, 2, 3, 4, 1, 1 ] ]
## gap> PowerMap( tbl, 137, 2 );
## 2
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "PowerMap", [ IsNearlyCharacterTable, IsInt ] );
DeclareOperation( "PowerMap", [ IsNearlyCharacterTable, IsInt, IsInt ] );
DeclareOperation( "PowerMapOp", [ IsNearlyCharacterTable, IsInt ] );
DeclareOperation( "PowerMapOp", [ IsNearlyCharacterTable, IsInt, IsInt ] );
DeclareAttributeSuppCT( "ComputedPowerMaps",
IsNearlyCharacterTable, "mutable", [ "class" ] );
#############################################################################
##
#O PossiblePowerMaps( <tbl>, <p>[, <options>] )
##
## <#GAPDoc Label="PossiblePowerMaps">
## <ManSection>
## <Oper Name="PossiblePowerMaps" Arg='tbl, p[, options]'/>
##
## <Description>
## For the ordinary character table <A>tbl</A> of a group <M>G</M>
## and a prime integer <A>p</A>,
## <Ref Oper="PossiblePowerMaps"/> returns the list of all maps that have
## the following properties of the <M>p</M>-th power map of <A>tbl</A>.
## (Representative orders are used only if the
## <Ref Attr="OrdersClassRepresentatives"/> value of <A>tbl</A> is known.
##
## <Enum>
## <Item>
## For class <M>i</M>, the centralizer order of the image is a multiple of
## the <M>i</M>-th centralizer order;
## if the elements in the <M>i</M>-th class have order coprime to <M>p</M>
## then the centralizer orders of class <M>i</M> and its image are equal.
## </Item>
## <Item>
## Let <M>n</M> be the order of elements in class <M>i</M>.
## If <A>prime</A> divides <M>n</M> then the images have order <M>n/p</M>;
## otherwise the images have order <M>n</M>.
## These criteria are checked in <Ref Func="InitPowerMap"/>.
## </Item>
## <Item>
## For each character <M>\chi</M> of <M>G</M> and each element <M>g</M>
## in <M>G</M>, the values <M>\chi(g^p)</M> and
## <C>GaloisCyc</C><M>( \chi(g), p )</M> are
## algebraic integers that are congruent modulo <M>p</M>;
## if <M>p</M> does not divide the element order of <M>g</M>
## then the two values are equal.
## This congruence is checked for the characters specified below in
## the discussion of the <A>options</A> argument;
## For linear characters <M>\lambda</M> among these characters,
## the condition <M>\chi(g)^p = \chi(g^p)</M> is checked.
## The corresponding function is
## <Ref Func="Congruences" Label="for character tables"/>.
## </Item>
## <Item>
## For each character <M>\chi</M> of <M>G</M>, the kernel is a normal
## subgroup <M>N</M>, and <M>g^p \in N</M> for all <M>g \in N</M>;
## moreover, if <M>N</M> has index <M>p</M> in <M>G</M> then
## <M>g^p \in N</M> for all <M>g \in G</M>,
## and if the index of <M>N</M> in <M>G</M> is coprime to <M>p</M> then
## <M>g^p \not \in N</M> for each <M>g \not \in N</M>.
## These conditions are checked for the kernels of all characters
## <M>\chi</M> specified below,
## the corresponding function is <Ref Func="ConsiderKernels"/>.
## </Item>
## <Item>
## If <M>p</M> is larger than the order <M>m</M> of an element
## <M>g \in G</M> then the class of <M>g^p</M> is determined by the power
## maps for primes dividing the residue of <M>p</M> modulo <M>m</M>.
## If these power maps are stored in the <Ref Attr="ComputedPowerMaps"/>
## value of <A>tbl</A> then this information is used.
## This criterion is checked in <Ref Func="ConsiderSmallerPowerMaps"/>.
## </Item>
## <Item>
## For each character <M>\chi</M> of <M>G</M>,
## the symmetrization <M>\psi</M> defined by
## <M>\psi(g) = (\chi(g)^p - \chi(g^p))/p</M> is a character.
## This condition is checked for the kernels of all characters
## <M>\chi</M> specified below,
## the corresponding function is
## <Ref Func="PowerMapsAllowedBySymmetrizations"/>.
## </Item>
## </Enum>
## <P/>
## If <A>tbl</A> is a Brauer table, the possibilities are computed
## from those for the underlying ordinary table.
## <P/>
## The optional argument <A>options</A>, if given, must be a record that may
## have the following components:
## <List>
## <Mark><C>chars</C>:</Mark>
## <Item>
## a list of characters which are used for the check of the criteria
## 3., 4., and 6.;
## the default is <C>Irr( <A>tbl</A> )</C>,
## </Item>
## <Mark><C>powermap</C>:</Mark>
## <Item>
## a parametrized map which is an approximation of the desired map
## </Item>
## <Mark><C>decompose</C>:</Mark>
## <Item>
## a Boolean;
## a <K>true</K> value indicates that all constituents of the
## symmetrizations of <C>chars</C> computed for criterion 6. lie in
## <C>chars</C>,
## so the symmetrizations can be decomposed into elements of <C>chars</C>;
## the default value of <C>decompose</C> is <K>true</K> if <C>chars</C>
## is not bound and <C>Irr( <A>tbl</A> )</C> is known,
## otherwise <K>false</K>,
## </Item>
## <Mark><C>quick</C>:</Mark>
## <Item>
## a Boolean;
## if <K>true</K> then the subroutines are called with value <K>true</K>
## for the argument <A>quick</A>;
## especially, as soon as only one candidate remains
## this candidate is returned immediately;
## the default value is <K>false</K>,
## </Item>
## <Mark><C>parameters</C>:</Mark>
## <Item>
## a record with components <C>maxamb</C>, <C>minamb</C> and <C>maxlen</C>
## which control the subroutine
## <Ref Func="PowerMapsAllowedBySymmetrizations"/>;
## it only uses characters with current indeterminateness up to
## <C>maxamb</C>,
## tests decomposability only for characters with current
## indeterminateness at least <C>minamb</C>,
## and admits a branch according to a character only if there is one
## with at most <C>maxlen</C> possible symmetrizations.
## </Item>
## </List>
## <P/>
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "U4(3).4" );;
## gap> PossiblePowerMaps( tbl, 2 );
## [ [ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, 6, 14, 9, 1, 1, 2, 2, 3, 4,
## 5, 6, 8, 9, 9, 10, 11, 12, 16, 16, 16, 16, 17, 17, 18, 18, 18,
## 18, 20, 20, 20, 20, 22, 22, 24, 24, 25, 26, 28, 28, 29, 29 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "PossiblePowerMaps", [ IsCharacterTable, IsInt ] );
DeclareOperation( "PossiblePowerMaps", [ IsCharacterTable, IsInt,
IsRecord ] );
#############################################################################
##
#F ElementOrdersPowerMap( <powermap> )
##
## <#GAPDoc Label="ElementOrdersPowerMap">
## <ManSection>
## <Func Name="ElementOrdersPowerMap" Arg='powermap'/>
##
## <Description>
## Let <A>powermap</A> be a nonempty list containing at position <M>p</M>,
## if bound, the <M>p</M>-th power map of a character table or group.
## <Ref Func="ElementOrdersPowerMap"/> returns a list of the same length as
## each entry in <A>powermap</A>, with entry at position <M>i</M> equal to
## the order of elements in class <M>i</M> if this order is uniquely
## determined by <A>powermap</A>,
## and equal to an unknown (see Chapter <Ref Chap="Unknowns"/>)
## otherwise.
## <P/>
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "U4(3).4" );;
## gap> known:= ComputedPowerMaps( tbl );;
## gap> Length( known );
## 7
## gap> sub:= ShallowCopy( known );; Unbind( sub[7] );
## gap> ElementOrdersPowerMap( sub );
## [ 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, Unknown(1), Unknown(2), 8, 9, 12, 2,
## 2, 4, 4, 6, 6, 6, 8, 10, 12, 12, 12, Unknown(3), Unknown(4), 4, 4,
## 4, 4, 4, 4, 8, 8, 8, 8, 12, 12, 12, 12, 12, 12, 20, 20, 24, 24,
## Unknown(5), Unknown(6), Unknown(7), Unknown(8) ]
## gap> ord:= ElementOrdersPowerMap( known );
## [ 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 12, 2, 2, 4, 4, 6, 6, 6,
## 8, 10, 12, 12, 12, 14, 14, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 12, 12,
## 12, 12, 12, 12, 20, 20, 24, 24, 28, 28, 28, 28 ]
## gap> ord = OrdersClassRepresentatives( tbl );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ElementOrdersPowerMap" );
#############################################################################
##
#F PowerMapByComposition( <tbl>, <n> ) . . for char. table and pos. integer
##
## <#GAPDoc Label="PowerMapByComposition">
## <ManSection>
## <Func Name="PowerMapByComposition" Arg='tbl, n'/>
##
## <Description>
## <A>tbl</A> must be a nearly character table,
## and <A>n</A> a positive integer.
## If the power maps for all prime divisors of <A>n</A> are stored in the
## <Ref Attr="ComputedPowerMaps"/> list of <A>tbl</A> then
## <Ref Func="PowerMapByComposition"/> returns
## the <A>n</A>-th power map of <A>tbl</A>.
## Otherwise <K>fail</K> is returned.
## <P/>
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "U4(3).4" );; exp:= Exponent( tbl );
## 2520
## gap> PowerMapByComposition( tbl, exp );
## [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
## 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
## 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
## gap> Length( ComputedPowerMaps( tbl ) );
## 7
## gap> PowerMapByComposition( tbl, 11 );
## fail
## gap> PowerMap( tbl, 11 );;
## gap> PowerMapByComposition( tbl, 11 );
## [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
## 20, 21, 22, 23, 24, 26, 25, 27, 28, 29, 31, 30, 33, 32, 35, 34, 37,
## 36, 39, 38, 41, 40, 43, 42, 45, 44, 47, 46, 49, 48, 51, 50, 53, 52 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PowerMapByComposition" );
#############################################################################
##
## <#GAPDoc Label="[3]{ctblmaps}">
## The permutation group of matrix automorphisms
## (see <Ref Oper="MatrixAutomorphisms"/>)
## acts on the possible power maps returned by
## <Ref Oper="PossiblePowerMaps"/>
## by permuting a list via <Ref Oper="Permuted"/>
## and then mapping the images via <Ref Func="OnPoints"/>.
## Note that by definition, the group of <E>table</E> automorphisms
## acts trivially.
## <#/GAPDoc>
##
#############################################################################
##
#F OrbitPowerMaps( <map>, <permgrp> )
##
## <#GAPDoc Label="OrbitPowerMaps">
## <ManSection>
## <Func Name="OrbitPowerMaps" Arg='map, permgrp'/>
##
## <Description>
## returns the orbit of the power map <A>map</A> under the action of the
## permutation group <A>permgrp</A>
## via a combination of <Ref Oper="Permuted"/> and <Ref Func="OnPoints"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "OrbitPowerMaps" );
#############################################################################
##
#F RepresentativesPowerMaps( <listofmaps>, <permgrp> )
##
## <#GAPDoc Label="RepresentativesPowerMaps">
## <ManSection>
## <Func Name="RepresentativesPowerMaps" Arg='listofmaps, permgrp'/>
##
## <Description>
## <Index>matrix automorphisms</Index>
## returns a list of orbit representatives of the power maps in the list
## <A>listofmaps</A> under the action of the permutation group
## <A>permgrp</A>
## via a combination of <Ref Oper="Permuted"/> and <Ref Func="OnPoints"/>.
## <P/>
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "3.McL" );;
## gap> grp:= MatrixAutomorphisms( Irr( tbl ) ); Size( grp );
## <permutation group with 5 generators>
## 32
## gap> poss:= PossiblePowerMaps( CharacterTable( "3.McL" ), 3 );
## [ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
## 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 9, 8, 37,
## 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
## 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ],
## [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
## 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37,
## 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
## 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]
## gap> reps:= RepresentativesPowerMaps( poss, grp );
## [ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
## 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37,
## 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
## 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]
## gap> orb:= OrbitPowerMaps( reps[1], grp );
## [ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
## 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37,
## 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
## 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ],
## [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
## 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 9, 8, 37,
## 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
## 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]
## gap> Parametrized( orb );
## [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
## 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, [ 8, 9 ],
## [ 8, 9 ], 37, 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52,
## 52, 49, 49, 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "RepresentativesPowerMaps" );
#############################################################################
##
## 3. Class Fusions between Character Tables
##
## <#GAPDoc Label="[4]{ctblmaps}">
## <Index>fusions</Index><Index>subgroup fusions</Index>
## For a group <M>G</M> and a subgroup <M>H</M> of <M>G</M>,
## the fusion map between the character table of <M>H</M> and the character
## table of <M>G</M> is represented by a list that stores at position
## <M>i</M> the position of the <M>i</M>-th class of the table of <M>H</M>
## in the classes list of the table of <M>G</M>.
## <P/>
## For ordinary character tables <A>tbl1</A> and <A>tbl2</A> of <M>H</M> and
## <M>G</M>, with access to the groups <M>H</M> and <M>G</M>,
## the class fusion between <A>tbl1</A> and <A>tbl2</A> can be computed
## using the identifications of the conjugacy classes of <M>H</M> with the
## classes of <A>tbl1</A> and the conjugacy classes of <M>G</M> with the
## classes of <A>tbl2</A>.
## For two ordinary character tables without access to an underlying group,
## or in the situation that the group stored in <A>tbl1</A> is not
## physically a subgroup of the group stored in <A>tbl2</A> but an
## isomorphic copy, in general the class fusion is not uniquely determined
## by the information stored on the tables such as irreducible characters
## and power maps.
## So only necessary conditions can be checked in this case,
## which in general yields only a list of several possibilities for the
## desired class fusion.
## Character tables of the &GAP; character table library store various
## class fusions that are regarded as important,
## for example fusions from maximal subgroups
## (see <Ref Attr="ComputedClassFusions"/>
## and <Ref Attr="Maxes" BookName="ctbllib"/> in the manual for the &GAP;
## Character Table Library).
## <P/>
## Class fusions between Brauer tables can be derived from the class fusions
## between the underlying ordinary tables.
## The class fusion from a Brauer table to the underlying ordinary table is
## stored when the Brauer table is constructed from the ordinary table,
## so no method is needed to compute such a fusion.
## <P/>
## For (computing and) accessing the class fusion between two character
## tables,
## <Ref Oper="FusionConjugacyClasses" Label="for two character tables"/>
## can be used;
## if the class fusion cannot be uniquely determined then
## <Ref Oper="FusionConjugacyClasses" Label="for two character tables"/>
## returns <K>fail</K>.
## <P/>
## The list of all possible class fusion between two tables in the sense
## that certain necessary conditions are satisfied can be computed with
## <Ref Oper="PossibleClassFusions"/>.
## This provides a default strategy, the subroutines are listed in
## Section <Ref Sect="Subroutines for the Construction of Class Fusions"/>.
## <P/>
## It should be noted that all the following functions except
## <Ref Oper="FusionConjugacyClasses" Label="for two character tables"/>
## deal only with the situation of class fusions from subgroups.
## The computation of <E>factor fusions</E> from a character table to the
## table of a factor group is not dealt with here.
## Since the ordinary character table of a group <M>G</M> determines the
## character tables of all factor groups of <M>G</M>, the factor fusion to a
## given character table of a factor group of <M>G</M> is determined up to
## table automorphisms (see <Ref Attr="AutomorphismsOfTable"/>) once
## the class positions of the kernel of the natural epimorphism have been
## fixed.
## <#/GAPDoc>
##
#############################################################################
##
#O FusionConjugacyClasses( <tbl1>, <tbl2> )
#O FusionConjugacyClasses( <H>, <G> )
#O FusionConjugacyClasses( <hom>[, <tbl1>, <tbl2>] )
#O FusionConjugacyClassesOp( <tbl1>, <tbl2> )
#A FusionConjugacyClassesOp( <hom> )
##
## <#GAPDoc Label="FusionConjugacyClasses">
## <ManSection>
## <Heading>FusionConjugacyClasses</Heading>
## <Oper Name="FusionConjugacyClasses" Arg='tbl1, tbl2'
## Label="for two character tables"/>
## <Oper Name="FusionConjugacyClasses" Arg='H, G'
## Label="for two groups"/>
## <Oper Name="FusionConjugacyClasses" Arg='hom[, tbl1, tbl2]'
## Label="for a homomorphism"/>
## <Oper Name="FusionConjugacyClassesOp" Arg='tbl1, tbl2'
## Label="for two character tables"/>
## <Attr Name="FusionConjugacyClassesOp" Arg='hom'
## Label="for a homomorphism"/>
##
## <Description>
## Called with two character tables <A>tbl1</A> and <A>tbl2</A>,
## <Ref Oper="FusionConjugacyClasses" Label="for two character tables"/>
## returns the fusion of conjugacy classes between <A>tbl1</A> and
## <A>tbl2</A>.
## (If one of the tables is a Brauer table,
## it will delegate this task to the underlying ordinary table.)
## <P/>
## Called with two groups <A>H</A> and <A>G</A> where <A>H</A> is a subgroup
## of <A>G</A>,
## <Ref Oper="FusionConjugacyClasses" Label="for two groups"/> returns
## the fusion of conjugacy classes between <A>H</A> and <A>G</A>.
## This is done by delegating to the ordinary character tables of <A>H</A>
## and <A>G</A>,
## since class fusions are stored only for character tables and not for
## groups.
## <P/>
## Note that the returned class fusion refers to the ordering of conjugacy
## classes in the character tables if the arguments are character tables
## and to the ordering of conjugacy classes in the groups if the arguments
## are groups
## (see <Ref Attr="ConjugacyClasses" Label="for character tables"/>).
## <P/>
## Called with a group homomorphism <A>hom</A>,
## <Ref Oper="FusionConjugacyClasses" Label="for a homomorphism"/> returns
## the fusion of conjugacy classes between the preimage and the image of
## <A>hom</A>;
## contrary to the two cases above,
## also factor fusions can be handled by this variant.
## If <A>hom</A> is the only argument then the class fusion refers to the
## ordering of conjugacy classes in the groups.
## If the character tables of preimage and image are given as <A>tbl1</A>
## and <A>tbl2</A>, respectively (each table with its group stored),
## then the fusion refers to the ordering of classes in these tables.
## <P/>
## If no class fusion exists or if the class fusion is not uniquely
## determined, <K>fail</K> is returned; this may happen when
## <Ref Oper="FusionConjugacyClasses" Label="for two character tables"/> is
## called with two character tables that do not know compatible underlying
## groups.
## <P/>
## Methods for the computation of class fusions can be installed for
## the operation
## <Ref Oper="FusionConjugacyClassesOp" Label="for two character tables"/>.
## <P/>
## <Example><![CDATA[
## gap> s4:= SymmetricGroup( 4 );
## Sym( [ 1 .. 4 ] )
## gap> tbls4:= CharacterTable( s4 );;
## gap> d8:= SylowSubgroup( s4, 2 );
## Group([ (1,2), (3,4), (1,3)(2,4) ])
## gap> FusionConjugacyClasses( d8, s4 );
## [ 1, 2, 3, 3, 5 ]
## gap> tbls5:= CharacterTable( "S5" );;
## gap> FusionConjugacyClasses( CharacterTable( "A5" ), tbls5 );
## [ 1, 2, 3, 4, 4 ]
## gap> FusionConjugacyClasses(CharacterTable("A5"), CharacterTable("J1"));
## fail
## gap> PossibleClassFusions(CharacterTable("A5"), CharacterTable("J1"));
## [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 5, 4 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "FusionConjugacyClasses",
[ IsNearlyCharacterTable, IsNearlyCharacterTable ] );
DeclareOperation( "FusionConjugacyClasses", [ IsGroup, IsGroup ] );
DeclareOperation( "FusionConjugacyClasses", [ IsGeneralMapping ] );
DeclareOperation( "FusionConjugacyClasses",
[ IsGeneralMapping, IsNearlyCharacterTable, IsNearlyCharacterTable ] );
DeclareAttribute( "FusionConjugacyClassesOp", IsGeneralMapping );
DeclareOperation( "FusionConjugacyClassesOp",
[ IsNearlyCharacterTable, IsNearlyCharacterTable ] );
DeclareOperation( "FusionConjugacyClassesOp",
[ IsGeneralMapping, IsNearlyCharacterTable, IsNearlyCharacterTable ] );
#############################################################################
##
#A ComputedClassFusions( <tbl> )
##
## <#GAPDoc Label="ComputedClassFusions">
## <ManSection>
## <Attr Name="ComputedClassFusions" Arg='tbl'/>
##
## <Description>
## The class fusions from the character table <A>tbl</A> that have been
## computed already by
## <Ref Oper="FusionConjugacyClasses" Label="for two character tables"/> or
## explicitly stored by <Ref Func="StoreFusion"/>
## are stored in the <Ref Attr="ComputedClassFusions"/> list of <A>tbl1</A>.
## Each entry of this list is a record with the following components.
##
## <List>
## <Mark><C>name</C></Mark>
## <Item>
## the <Ref Attr="Identifier" Label="for character tables"/> value
## of the character table to which the fusion maps,
## </Item>
## <Mark><C>map</C></Mark>
## <Item>
## the list of positions of image classes,
## </Item>
## <Mark><C>text</C> (optional)</Mark>
## <Item>
## a string giving additional information about the fusion map,
## for example whether the map is uniquely determined by the character
## tables,
## </Item>
## <Mark><C>specification</C> (optional, rarely used)</Mark>
## <Item>
## a value that distinguishes different fusions between the same tables.
## </Item>
## </List>
## <P/>
## Note that stored fusion maps may differ from the maps returned by
## <Ref Func="GetFusionMap"/> and the maps entered by
## <Ref Func="StoreFusion"/> if the table <A>destination</A> has a
## nonidentity <Ref Attr="ClassPermutation"/> value.
## So if one fetches a fusion map from a table <A>tbl1</A> to a table
## <A>tbl2</A> via access to the data in the
## <Ref Attr="ComputedClassFusions"/> list of <A>tbl1</A> then the stored
## value must be composed with the <Ref Attr="ClassPermutation"/> value of
## <A>tbl2</A> in order to obtain the correct class fusion.
## (If one handles fusions only via <Ref Func="GetFusionMap"/> and
## <Ref Func="StoreFusion"/> then this adjustment is made automatically.)
## <P/>
## Fusions are identified via the
## <Ref Attr="Identifier" Label="for character tables"/> value of the
## destination table and not by this table itself because many fusions
## between character tables in the &GAP; character table library are stored
## on library tables,
## and it is not desirable to load together with a library table also all
## those character tables that occur as destinations of fusions from this
## table.
## <P/>
## For storing fusions and accessing stored fusions,
## see also <Ref Func="GetFusionMap"/>, <Ref Func="StoreFusion"/>.
## For accessing the identifiers of tables that store a fusion into a
## given character table, see <Ref Attr="NamesOfFusionSources"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttributeSuppCT( "ComputedClassFusions",
IsNearlyCharacterTable, "mutable", [ "class" ] );
#############################################################################
##
#F GetFusionMap( <source>, <destination>[, <specification>] )
##
## <#GAPDoc Label="GetFusionMap">
## <ManSection>
## <Func Name="GetFusionMap" Arg='source, destination[, specification]'/>
##
## <Description>
## For two ordinary character tables <A>source</A> and <A>destination</A>,
## <Ref Func="GetFusionMap"/> checks whether the
## <Ref Attr="ComputedClassFusions"/> list of <A>source</A>
## contains a record with <C>name</C> component
## <C>Identifier( <A>destination</A> )</C>,
## and returns the <C>map</C> component of the first such record.
## <C>GetFusionMap( <A>source</A>, <A>destination</A>,
## <A>specification</A> )</C> fetches
## that fusion map for which the record additionally has the
## <C>specification</C> component <A>specification</A>.
## <P/>
## If both <A>source</A> and <A>destination</A> are Brauer tables,
## first the same is done, and if no fusion map was found then
## <Ref Func="GetFusionMap"/> looks whether a fusion map between the
## ordinary tables is stored;
## if so then the fusion map between <A>source</A> and <A>destination</A>
## is stored on <A>source</A>, and then returned.
## <P/>
## If no appropriate fusion is found, <Ref Func="GetFusionMap"/> returns
## <K>fail</K>.
## For the computation of class fusions, see
## <Ref Oper="FusionConjugacyClasses" Label="for two character tables"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "GetFusionMap" );
#############################################################################
##
#F StoreFusion( <source>, <fusion>, <destination> )
##
## <#GAPDoc Label="StoreFusion">
## <ManSection>
## <Func Name="StoreFusion" Arg='source, fusion, destination'/>
##
## <Description>
## For two character tables <A>source</A> and <A>destination</A>,
## <Ref Func="StoreFusion"/> stores the fusion <A>fusion</A> from
## <A>source</A> to <A>destination</A> in the
## <Ref Attr="ComputedClassFusions"/> list of <A>source</A>,
## and adds the <Ref Attr="Identifier" Label="for character tables"/> string
## of <A>destination</A> to the <Ref Attr="NamesOfFusionSources"/> list of
## <A>destination</A>.
## <P/>
## <A>fusion</A> can either be a fusion map (that is, the list of positions
## of the image classes) or a record as described
## in <Ref Attr="ComputedClassFusions"/>.
## <P/>
## If fusions to <A>destination</A> are already stored on <A>source</A> then
## another fusion can be stored only if it has a record component
## <C>specification</C> that distinguishes it from the stored fusions.
## In the case of such an ambiguity, <Ref Func="StoreFusion"/> raises an
## error.
## <P/>
## <Example><![CDATA[
## gap> tbld8:= CharacterTable( d8 );;
## gap> ComputedClassFusions( tbld8 );
## [ rec( map := [ 1, 2, 3, 3, 5 ], name := "CT1" ) ]
## gap> Identifier( tbls4 );
## "CT1"
## gap> GetFusionMap( tbld8, tbls4 );
## [ 1, 2, 3, 3, 5 ]
## gap> GetFusionMap( tbls4, tbls5 );
## fail
## gap> poss:= PossibleClassFusions( tbls4, tbls5 );
## [ [ 1, 5, 2, 3, 6 ] ]
## gap> StoreFusion( tbls4, poss[1], tbls5 );
## gap> GetFusionMap( tbls4, tbls5 );
## [ 1, 5, 2, 3, 6 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "StoreFusion" );
#############################################################################
##
#A NamesOfFusionSources( <tbl> )
##
## <#GAPDoc Label="NamesOfFusionSources">
## <ManSection>
## <Attr Name="NamesOfFusionSources" Arg='tbl'/>
##
## <Description>
## For a character table <A>tbl</A>,
## <Ref Attr="NamesOfFusionSources"/> returns the list of identifiers of all
## those character tables that are known to have fusions to <A>tbl</A>
## stored.
## The <Ref Attr="NamesOfFusionSources"/> value is updated whenever a fusion
## to <A>tbl</A> is stored using <Ref Func="StoreFusion"/>.
## <P/>
## <Example><![CDATA[
## gap> NamesOfFusionSources( tbls4 );
## [ "CT2" ]
## gap> Identifier( CharacterTable( d8 ) );
## "CT2"
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttributeSuppCT( "NamesOfFusionSources",
IsNearlyCharacterTable, "mutable", [] );
#############################################################################
##
#O PossibleClassFusions( <subtbl>, <tbl>[, <options>] )
##
## <#GAPDoc Label="PossibleClassFusions">
## <ManSection>
## <Oper Name="PossibleClassFusions" Arg='subtbl, tbl[, options]'/>
##
## <Description>
## For two ordinary character tables <A>subtbl</A> and <A>tbl</A> of the
## groups <M>H</M> and <M>G</M>,
## <Ref Oper="PossibleClassFusions"/> returns the list of all maps that have
## the following properties of class fusions from <A>subtbl</A> to
## <A>tbl</A>.
##
## <Enum>
## <Item>
## For class <M>i</M>, the centralizer order of the image in <M>G</M> is a
## multiple of the <M>i</M>-th centralizer order in <M>H</M>,
## and the element orders in the <M>i</M>-th class and its image are
## equal.
## These criteria are checked in <Ref Func="InitFusion"/>.
## </Item>
## <Item>
## The class fusion commutes with power maps.
## This is checked using <Ref Func="TestConsistencyMaps"/>.
## </Item>
## <Item>
## If the permutation character of <M>G</M> corresponding to the action of
## <M>G</M> on the cosets of <M>H</M> is specified (see the discussion of
## the <A>options</A> argument below)
## then it prescribes for each class <M>C</M> of
## <M>G</M> the number of elements of <M>H</M> fusing into <M>C</M>.
## The corresponding function is <Ref Func="CheckPermChar"/>.
## </Item>
## <Item>
## The table automorphisms of <A>tbl</A>
## (see <Ref Attr="AutomorphismsOfTable"/>) are
## used in order to compute only orbit representatives.
## (But note that the list returned by <Ref Oper="PossibleClassFusions"/>
## contains the full orbits.)
## </Item>
## <Item>
## For each character <M>\chi</M> of <M>G</M>, the restriction to <M>H</M>
## via the class fusion is a character of <M>H</M>.
## This condition is checked for all characters specified below,
## the corresponding function is
## <Ref Func="FusionsAllowedByRestrictions"/>.
## </Item>
## <Item>
## The class multiplication coefficients in <A>subtbl</A> do not exceed
## the corresponding coefficients in <A>tbl</A>.
## This is checked in <Ref Func="ConsiderStructureConstants"/>,
## see also the comment on the parameter <C>verify</C> below.
## </Item>
## </Enum>
## <P/>
## If <A>subtbl</A> and <A>tbl</A> are Brauer tables then the possibilities
## are computed from those for the underlying ordinary tables.
## <P/>
## The optional argument <A>options</A> must be a record that may have the
## following components:
##
## <List>
## <Mark><C>chars</C></Mark>
## <Item>
## a list of characters of <A>tbl</A> which are used for the check
## of 5.; the default is <C>Irr( <A>tbl</A> )</C>,
## </Item>
## <Mark><C>subchars</C></Mark>
## <Item>
## a list of characters of <A>subtbl</A> which are constituents of the
## restrictions of <C>chars</C>,
## the default is <C>Irr( <A>subtbl</A> )</C>,
## </Item>
## <Mark><C>fusionmap</C></Mark>
## <Item>
## a parametrized map which is an approximation of the desired map,
## </Item>
## <Mark><C>decompose</C></Mark>
## <Item>
## a Boolean;
## a <K>true</K> value indicates that all constituents of the restrictions
## of <C>chars</C> computed for criterion 5. lie in <C>subchars</C>,
## so the restrictions can be decomposed into elements of <C>subchars</C>;
## the default value of <C>decompose</C> is <K>true</K> if <C>subchars</C>
## is not bound and <C>Irr( <A>subtbl</A> )</C> is known,
## otherwise <K>false</K>,
## </Item>
## <Mark><C>permchar</C></Mark>
## <Item>
## (a values list of) a permutation character; only those fusions
## affording that permutation character are computed,
## </Item>
## <Mark><C>quick</C></Mark>
## <Item>
## a Boolean;
## if <K>true</K> then the subroutines are called with value <K>true</K>
## for the argument <A>quick</A>;
## especially, as soon as only one possibility remains
## then this possibility is returned immediately;
## the default value is <K>false</K>
## (note that in situations where the group of <A>tbl</A> has no subgroups
## with character table <A>subtbl</A>, it may happen that setting
## <C>quick</C> to <K>true</K> causes <Ref Oper="PossibleClassFusions"/>
## to return solutions,
## whereas the value <K>false</K> yields an empty list),
## </Item>
## <Mark><C>verify</C></Mark>
## <Item>
## a Boolean;
## if <K>false</K> then <Ref Func="ConsiderStructureConstants"/> is called
## only if more than one orbit of possible class fusions exists,
## under the action of the groups of table automorphisms;
## the default value is <K>false</K> (because the computation of the
## structure constants is usually very time consuming, compared with
## checking the other criteria),
## </Item>
## <Mark><C>parameters</C></Mark>
## <Item>
## a record with components <C>maxamb</C>, <C>minamb</C> and <C>maxlen</C>
## (and perhaps some optional components) which control the subroutine
## <Ref Func="FusionsAllowedByRestrictions"/>;
## it only uses characters with current indeterminateness up to
## <C>maxamb</C>,
## tests decomposability only for characters with current
## indeterminateness at least <C>minamb</C>,
## and admits a branch according to a character only if there is one
## with at most <C>maxlen</C> possible restrictions.
## </Item>
## </List>
## <P/>
## <Example><![CDATA[
## gap> subtbl:= CharacterTable( "U3(3)" );; tbl:= CharacterTable( "J4" );;
## gap> PossibleClassFusions( subtbl, tbl );
## [ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ],
## [ 1, 2, 4, 4, 5, 5, 6, 10, 13, 12, 14, 14, 21, 21 ],
## [ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 15, 15, 22, 22 ],
## [ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 16, 16, 22, 22 ],
## [ 1, 2, 4, 4, 6, 6, 6, 10, 13, 12, 15, 15, 22, 22 ],
## [ 1, 2, 4, 4, 6, 6, 6, 10, 13, 12, 16, 16, 22, 22 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "PossibleClassFusions",
[ IsNearlyCharacterTable, IsNearlyCharacterTable ] );
DeclareOperation( "PossibleClassFusions",
[ IsNearlyCharacterTable, IsNearlyCharacterTable, IsRecord ] );
#############################################################################
##
## <#GAPDoc Label="[5]{ctblmaps}">
## The permutation groups of table automorphisms
## (see <Ref Attr="AutomorphismsOfTable"/>)
## of the subgroup table <A>subtbl</A> and the supergroup table <A>tbl</A>
## act on the possible class fusions from <A>subtbl</A> to <A>tbl</A>
## that are returned by <Ref Oper="PossibleClassFusions"/>,
## the former by permuting a list via <Ref Oper="Permuted"/>,
## the latter by mapping the images via <Ref Func="OnPoints"/>.
## <P/>
## If a set of possible fusions with certain properties was computed
## that are not invariant under the full groups of table automorphisms
## then only a smaller group acts on this set.
## This may happen for example if a permutation character or if an explicit
## approximation of the fusion map was prescribed in the call of
## <Ref Oper="PossibleClassFusions"/>.
## <#/GAPDoc>
##
#############################################################################
##
#F OrbitFusions( <subtblautomorphisms>, <fusionmap>, <tblautomorphisms> )
##
## <#GAPDoc Label="OrbitFusions">
## <ManSection>
## <Func Name="OrbitFusions"
## Arg='subtblautomorphisms, fusionmap, tblautomorphisms'/>
##
## <Description>
## returns the orbit of the class fusion map <A>fusionmap</A> under the
## actions of the permutation groups <A>subtblautomorphisms</A> and
## <A>tblautomorphisms</A> of automorphisms of the character table of the
## subgroup and the supergroup, respectively.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "OrbitFusions" );
#############################################################################
##
#F RepresentativesFusions( <subtbl>, <listofmaps>, <tbl> )
##
## <#GAPDoc Label="RepresentativesFusions">
## <ManSection>
## <Func Name="RepresentativesFusions" Arg='subtbl, listofmaps, tbl'/>
##
## <Description>
## <Index>table automorphisms</Index>
## Let <A>listofmaps</A> be a list of class fusions from the character table
## <A>subtbl</A> to the character table <A>tbl</A>.
## <Ref Func="RepresentativesFusions"/> returns a list of orbit
## representatives of the class fusions under the action of maximal
## admissible subgroups of the table automorphism groups of these character
## tables.
## <P/>
## Instead of the character tables <A>subtbl</A> and <A>tbl</A>,
## also the permutation groups of their table automorphisms
## (see <Ref Attr="AutomorphismsOfTable"/>) may be entered.
## <P/>
## <Example><![CDATA[
## gap> fus:= GetFusionMap( subtbl, tbl );
## [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ]
## gap> orb:= OrbitFusions( AutomorphismsOfTable( subtbl ), fus,
## > AutomorphismsOfTable( tbl ) );
## [ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ],
## [ 1, 2, 4, 4, 5, 5, 6, 10, 13, 12, 14, 14, 21, 21 ] ]
## gap> rep:= RepresentativesFusions( subtbl, orb, tbl );
## [ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "RepresentativesFusions" );
#############################################################################
##
## 4. Utilities for Parametrized Maps
##
## <#GAPDoc Label="[6]{ctblmaps}">
## <Index Subkey="parametrized">map</Index>
## <Index>class functions</Index>
## A <E>parametrized map</E> is a list whose <M>i</M>-th entry is either
## unbound (which means that nothing is known about the image(s) of the
## <M>i</M>-th class) or the image of the <M>i</M>-th class
## (i.e., an integer for fusion maps, power maps, element orders etc.,
## and a cyclotomic for characters),
## or a list of possible images of the <M>i</M>-th class.
## In this sense, maps are special parametrized maps.
## We often identify a parametrized map <A>paramap</A> with the set of all
## maps <A>map</A> with the property that either
## <C><A>map</A>[i] = <A>paramap</A>[i]</C> or
## <C><A>map</A>[i]</C> is contained in the list <C><A>paramap</A>[i]</C>;
## we say then that <A>map</A> is contained in <A>paramap</A>.
## <P/>
## This definition implies that parametrized maps cannot be used to describe
## sets of maps where lists are possible images.
## An exception are strings which naturally arise as images when class names
## are considered.
## So strings and lists of strings are allowed in parametrized maps,
## and character constants
## (see Chapter <Ref Chap="Strings and Characters"/>)
## are not allowed in maps.
## <#/GAPDoc>
##
#############################################################################
##
#F CompositionMaps( <paramap2>, <paramap1>[, <class>] )
##
## <#GAPDoc Label="CompositionMaps">
## <ManSection>
## <Func Name="CompositionMaps" Arg='paramap2, paramap1[, class]'/>
##
## <Description>
## The composition of two parametrized maps <A>paramap1</A>, <A>paramap2</A>
## is defined as the parametrized map <A>comp</A> that contains
## all compositions <M>f_2 \circ f_1</M> of elements <M>f_1</M> of
## <A>paramap1</A> and <M>f_2</M> of <A>paramap2</A>.
## For example, the composition of a character <M>\chi</M> of a group
## <M>G</M> by a parametrized class fusion map from a subgroup <M>H</M> to
## <M>G</M> is the parametrized map that contains all restrictions of
## <M>\chi</M> by elements of the parametrized fusion map.
## <P/>
## <C>CompositionMaps(<A>paramap2</A>, <A>paramap1</A>)</C>
## is a parametrized map with entry
## <C>CompositionMaps(<A>paramap2</A>, <A>paramap1</A>, <A>class</A>)</C>
## at position <A>class</A>.
## If <C><A>paramap1</A>[<A>class</A>]</C> is an integer then
## <C>CompositionMaps(<A>paramap2</A>, <A>paramap1</A>, <A>class</A>)</C>
## is equal to <C><A>paramap2</A>[ <A>paramap1</A>[ <A>class</A> ] ]</C>.
## Otherwise it is the union of <C><A>paramap2</A>[<A>i</A>]</C> for
## <A>i</A> in <C><A>paramap1</A>[ <A>class</A> ]</C>.
## <P/>
## <Example><![CDATA[
## gap> map1:= [ 1, [ 2 .. 4 ], [ 4, 5 ], 1 ];;
## gap> map2:= [ [ 1, 2 ], 2, 2, 3, 3 ];;
## gap> CompositionMaps( map2, map1 );
## [ [ 1, 2 ], [ 2, 3 ], 3, [ 1, 2 ] ]
## gap> CompositionMaps( map1, map2 );
## [ [ 1, 2, 3, 4 ], [ 2 .. 4 ], [ 2 .. 4 ], [ 4, 5 ], [ 4, 5 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CompositionMaps" );
#############################################################################
##
#F InverseMap( <paramap> ) . . . . . . . . . . inverse of a parametrized map
##
## <#GAPDoc Label="InverseMap">
## <ManSection>
## <Func Name="InverseMap" Arg='paramap'/>
##
## <Description>
## For a parametrized map <A>paramap</A>,
## <Ref Func="InverseMap"/> returns a mutable parametrized map whose
## <M>i</M>-th entry is unbound if <M>i</M> is not in the image of
## <A>paramap</A>, equal to <M>j</M> if <M>i</M> is (in) the image of
## <C><A>paramap</A>[<A>j</A>]</C> exactly for <M>j</M>,
## and equal to the set of all preimages of <M>i</M> under <A>paramap</A>
## otherwise.
## <P/>
## We have
## <C>CompositionMaps( <A>paramap</A>, InverseMap( <A>paramap</A> ) )</C>
## the identity map.
## <P/>
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "2.A5" );; f:= CharacterTable( "A5" );;
## gap> fus:= GetFusionMap( tbl, f );
## [ 1, 1, 2, 3, 3, 4, 4, 5, 5 ]
## gap> inv:= InverseMap( fus );
## [ [ 1, 2 ], 3, [ 4, 5 ], [ 6, 7 ], [ 8, 9 ] ]
## gap> CompositionMaps( fus, inv );
## [ 1, 2, 3, 4, 5 ]
## gap> # transfer a power map ``up'' to the factor group
## gap> pow:= PowerMap( tbl, 2 );
## [ 1, 1, 2, 4, 4, 8, 8, 6, 6 ]
## gap> CompositionMaps( fus, CompositionMaps( pow, inv ) );
## [ 1, 1, 3, 5, 4 ]
## gap> last = PowerMap( f, 2 );
## true
## gap> # transfer a power map of the factor group ``down'' to the group
## gap> CompositionMaps( inv, CompositionMaps( PowerMap( f, 2 ), fus ) );
## [ [ 1, 2 ], [ 1, 2 ], [ 1, 2 ], [ 4, 5 ], [ 4, 5 ], [ 8, 9 ],
## [ 8, 9 ], [ 6, 7 ], [ 6, 7 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "InverseMap" );
#############################################################################
##
#F ProjectionMap( <fusionmap> ) . . . . projection corresp. to a fusion map
##
## <#GAPDoc Label="ProjectionMap">
## <ManSection>
## <Func Name="ProjectionMap" Arg='fusionmap'/>
##
## <Description>
## For a map <A>fusionmap</A>,
## <Ref Func="ProjectionMap"/> returns a parametrized map
## whose <M>i</M>-th entry is unbound if <M>i</M> is not in the image of
## <A>fusionmap</A>,
## and equal to <M>j</M> if <M>j</M> is the smallest position such that
## <M>i</M> is the image of <A>fusionmap</A><C>[</C><M>j</M><C>]</C>.
## <P/>
## We have
## <C>CompositionMaps( <A>fusionmap</A>, ProjectionMap( <A>fusionmap</A> ) )</C>
## the identity map, i.e., first projecting and then fusing yields the
## identity.
## Note that <A>fusionmap</A> must <E>not</E> be a parametrized map.
## <P/>
## <Example><![CDATA[
## gap> ProjectionMap( [ 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 5, 6, 6, 6 ] );
## [ 1, 4, 7, 8, 9, 12 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ProjectionMap" );
#############################################################################
##
#F Indirected( <character>, <paramap> )
##
## <#GAPDoc Label="Indirected">
## <ManSection>
## <Func Name="Indirected" Arg='character, paramap'/>
##
## <Description>
## For a map <A>character</A> and a parametrized map <A>paramap</A>,
## <Ref Func="Indirected"/> returns a parametrized map whose entry at
## position <M>i</M> is
## <A>character</A><C>[ </C><A>paramap</A><C>[</C><M>i</M><C>] ]</C>
## if <A>paramap</A><C>[</C><M>i</M><C>]</C> is an integer,
## and an unknown (see Chapter <Ref Chap="Unknowns"/>) otherwise.
## <P/>
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "M12" );;
## gap> fus:= [ 1, 3, 4, [ 6, 7 ], 8, 10, [ 11, 12 ], [ 11, 12 ],
## > [ 14, 15 ], [ 14, 15 ] ];;
## gap> List( Irr( tbl ){ [ 1 .. 6 ] }, x -> Indirected( x, fus ) );
## [ [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
## [ 11, 3, 2, Unknown(9), 1, 0, Unknown(10), Unknown(11), 0, 0 ],
## [ 11, 3, 2, Unknown(12), 1, 0, Unknown(13), Unknown(14), 0, 0 ],
## [ 16, 0, -2, 0, 1, 0, 0, 0, Unknown(15), Unknown(16) ],
## [ 16, 0, -2, 0, 1, 0, 0, 0, Unknown(17), Unknown(18) ],
## [ 45, -3, 0, 1, 0, 0, -1, -1, 1, 1 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "Indirected" );
#############################################################################
##
#F Parametrized( <list> )
##
## <#GAPDoc Label="Parametrized">
## <ManSection>
## <Func Name="Parametrized" Arg='list'/>
##
## <Description>
## For a list <A>list</A> of (parametrized) maps of the same length,
## <Ref Func="Parametrized"/> returns the smallest parametrized map
## containing all elements of <A>list</A>.
## <P/>
## <Ref Func="Parametrized"/> is the inverse function to
## <Ref Func="ContainedMaps"/>.
## <P/>
## <Example><![CDATA[
## gap> Parametrized( [ [ 1, 2, 3, 4, 5 ], [ 1, 3, 2, 4, 5 ],
## > [ 1, 2, 3, 4, 6 ] ] );
## [ 1, [ 2, 3 ], [ 2, 3 ], 4, [ 5, 6 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "Parametrized" );
#############################################################################
##
#F ContainedMaps( <paramap> )
##
## <#GAPDoc Label="ContainedMaps">
## <ManSection>
## <Func Name="ContainedMaps" Arg='paramap'/>
##
## <Description>
## For a parametrized map <A>paramap</A>,
## <Ref Func="ContainedMaps"/> returns the set of all
## maps contained in <A>paramap</A>.
## <P/>
## <Ref Func="ContainedMaps"/> is the inverse function to
## <Ref Func="Parametrized"/> in the sense that
## <C>Parametrized( ContainedMaps( <A>paramap</A> ) )</C>
## is equal to <A>paramap</A>.
## <P/>
## <Example><![CDATA[
## gap> ContainedMaps( [ 1, [ 2, 3 ], [ 2, 3 ], 4, [ 5, 6 ] ] );
## [ [ 1, 2, 2, 4, 5 ], [ 1, 2, 2, 4, 6 ], [ 1, 2, 3, 4, 5 ],
## [ 1, 2, 3, 4, 6 ], [ 1, 3, 2, 4, 5 ], [ 1, 3, 2, 4, 6 ],
## [ 1, 3, 3, 4, 5 ], [ 1, 3, 3, 4, 6 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ContainedMaps" );
#############################################################################
##
#F UpdateMap( <character>, <paramap>, <indirected> )
##
## <#GAPDoc Label="UpdateMap">
## <ManSection>
## <Func Name="UpdateMap" Arg='character, paramap, indirected'/>
##
## <Description>
## Let <A>character</A> be a map, <A>paramap</A> a parametrized map,
## and <A>indirected</A> a parametrized map that is contained in
## <C>CompositionMaps( <A>character</A>, <A>paramap</A> )</C>.
## <P/>
## Then <Ref Func="UpdateMap"/> changes <A>paramap</A> to the parametrized
## map containing exactly the maps whose composition with <A>character</A>
## is equal to <A>indirected</A>.
## <P/>
## If a contradiction is detected then <K>false</K> is returned immediately,
## otherwise <K>true</K>.
## <P/>
## <Example><![CDATA[
## gap> subtbl:= CharacterTable("S4(4).2");; tbl:= CharacterTable("He");;
## gap> fus:= InitFusion( subtbl, tbl );;
## gap> fus;
## [ 1, 2, 2, [ 2, 3 ], 4, 4, [ 7, 8 ], [ 7, 8 ], 9, 9, 9, [ 10, 11 ],
## [ 10, 11 ], 18, 18, 25, 25, [ 26, 27 ], [ 26, 27 ], 2, [ 6, 7 ],
## [ 6, 7 ], [ 6, 7, 8 ], 10, 10, 17, 17, 18, [ 19, 20 ], [ 19, 20 ] ]
## gap> chi:= Irr( tbl )[2];
## Character( CharacterTable( "He" ), [ 51, 11, 3, 6, 0, 3, 3, -1, 1, 2,
## 0, 3*E(7)+3*E(7)^2+3*E(7)^4, 3*E(7)^3+3*E(7)^5+3*E(7)^6, 2,
## E(7)+E(7)^2+2*E(7)^3+E(7)^4+2*E(7)^5+2*E(7)^6,
## 2*E(7)+2*E(7)^2+E(7)^3+2*E(7)^4+E(7)^5+E(7)^6, 1, 1, 0, 0,
## -E(7)-E(7)^2-E(7)^4, -E(7)^3-E(7)^5-E(7)^6, E(7)+E(7)^2+E(7)^4,
## E(7)^3+E(7)^5+E(7)^6, 1, 0, 0, -1, -1, 0, 0, E(7)+E(7)^2+E(7)^4,
## E(7)^3+E(7)^5+E(7)^6 ] )
## gap> filt:= Filtered( Irr( subtbl ), x -> x[1] = 50 );
## [ Character( CharacterTable( "S4(4).2" ),
## [ 50, 10, 10, 2, 5, 5, -2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1,
## 10, 2, 2, 2, 1, 1, 0, 0, 0, -1, -1 ] ),
## Character( CharacterTable( "S4(4).2" ),
## [ 50, 10, 10, 2, 5, 5, -2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1,
## -10, -2, -2, -2, -1, -1, 0, 0, 0, 1, 1 ] ) ]
## gap> UpdateMap( chi, fus, filt[1] + TrivialCharacter( subtbl ) );
## true
## gap> fus;
## [ 1, 2, 2, 3, 4, 4, 8, 7, 9, 9, 9, 10, 10, 18, 18, 25, 25,
## [ 26, 27 ], [ 26, 27 ], 2, [ 6, 7 ], [ 6, 7 ], [ 6, 7 ], 10, 10,
## 17, 17, 18, [ 19, 20 ], [ 19, 20 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "UpdateMap" );
#############################################################################
##
#F MeetMaps( <paramap1>, <paramap2> )
##
## <#GAPDoc Label="MeetMaps">
## <ManSection>
## <Func Name="MeetMaps" Arg='paramap1, paramap2'/>
##
## <Description>
## For two parametrized maps <A>paramap1</A> and <A>paramap2</A>,
## <Ref Func="MeetMaps"/> changes <A>paramap1</A> such that the image of
## class <M>i</M> is the intersection of
## <A>paramap1</A><C>[</C><M>i</M><C>]</C>
## and <A>paramap2</A><C>[</C><M>i</M><C>]</C>.
## <P/>
## If this implies that no images remain for a class, the position of such a
## class is returned.
## If no such inconsistency occurs,
## <Ref Func="MeetMaps"/> returns <K>true</K>.
## <P/>
## <Example><![CDATA[
## gap> map1:= [ [ 1, 2 ], [ 3, 4 ], 5, 6, [ 7, 8, 9 ] ];;
## gap> map2:= [ [ 1, 3 ], [ 3, 4 ], [ 5, 6 ], 6, [ 8, 9, 10 ] ];;
## gap> MeetMaps( map1, map2 ); map1;
## true
## [ 1, [ 3, 4 ], 5, 6, [ 8, 9 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MeetMaps" );
#############################################################################
##
#F ImproveMaps( <map2>, <map1>, <composition>, <class> )
##
## <ManSection>
## <Func Name="ImproveMaps" Arg='map2, map1, composition, class'/>
##
## <Description>
## <Ref Func="ImproveMaps"/> is a utility for
## <Ref Func="CommutativeDiagram"/> and <Ref Func="TestConsistencyMaps"/>.
## <P/>
## <A>composition</A> must be a set that is known to be an upper bound for
## the composition <M>( <A>map2</A> \circ <A>map1</A> )[ <A>class</A> ]</M>.
## If <C><A>map1</A>[ <A>class</A> ]</C><M> = x</M> is unique then
## <M><A>map2</A>[ x ]</M> must be a set,
## it will be replaced by its intersection with <A>composition</A>;
## if <A>map1</A>[ <A>class</A> ] is a set then all elements <C>x</C> with
## empty <C>Intersection( <A>map2</A>[ x ], <A>composition</A> )</C>
## are excluded.
## <P/>
## <Ref Func="ImproveMaps"/> returns
## <List>
## <Mark>0</Mark>
## <Item>
## if no improvement was found,
## </Item>
## <Mark>-1</Mark>
## <Item>
## if <A>map1</A>[ <A>class</A> ] was improved,
## </Item>
## <Mark><A>x</A></Mark>
## <Item>
## if <A>map2</A>[ <A>x</A> ] was improved.
## </Item>
## </List>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "ImproveMaps" );
#############################################################################
##
#F CommutativeDiagram( <paramap1>, <paramap2>, <paramap3>, <paramap4>[,
#F <improvements>] )
##
## <#GAPDoc Label="CommutativeDiagram">
## <ManSection>
## <Func Name="CommutativeDiagram"
## Arg='paramap1, paramap2, paramap3, paramap4[, improvements]'/>
##
## <Description>
## Let <A>paramap1</A>, <A>paramap2</A>, <A>paramap3</A>, <A>paramap4</A> be
## parametrized maps covering parametrized maps <M>f_1</M>, <M>f_2</M>,
## <M>f_3</M>, <M>f_4</M> with the property
## that <C>CompositionMaps</C><M>( f_2, f_1 )</M> is equal to
## <C>CompositionMaps</C><M>( f_4, f_3 )</M>.
## <P/>
## <Ref Func="CommutativeDiagram"/> checks this consistency,
## and changes the arguments such that all possible images are removed that
## cannot occur in the parametrized maps <M>f_i</M>.
## <P/>
## The return value is <K>fail</K> if an inconsistency was found.
## Otherwise a record with the components <C>imp1</C>, <C>imp2</C>,
## <C>imp3</C>, <C>imp4</C> is returned, each bound to the list of positions
## where the corresponding parametrized map was changed,
## <P/>
## The optional argument <A>improvements</A> must be a record with
## components <C>imp1</C>, <C>imp2</C>, <C>imp3</C>, <C>imp4</C>.
## If such a record is specified then only diagrams are considered where
## entries of the <M>i</M>-th component occur as preimages of the
## <M>i</M>-th parametrized map.
## <P/>
## When an inconsistency is detected,
## <Ref Func="CommutativeDiagram"/> immediately returns <K>fail</K>.
## Otherwise a record is returned that contains four lists <C>imp1</C>,
## <M>\ldots</M>, <C>imp4</C>:
## The <M>i</M>-th component is the list of classes where the <M>i</M>-th
## argument was changed.
## <P/>
## <Example><![CDATA[
## gap> map1:= [[ 1, 2, 3 ], [ 1, 3 ]];; map2:= [[ 1, 2 ], 1, [ 1, 3 ]];;
## gap> map3:= [ [ 2, 3 ], 3 ];; map4:= [ , 1, 2, [ 1, 2 ] ];;
## gap> imp:= CommutativeDiagram( map1, map2, map3, map4 );
## rec( imp1 := [ 2 ], imp2 := [ 1 ], imp3 := [ ], imp4 := [ ] )
## gap> map1; map2; map3; map4;
## [ [ 1, 2, 3 ], 1 ]
## [ 2, 1, [ 1, 3 ] ]
## [ [ 2, 3 ], 3 ]
## [ , 1, 2, [ 1, 2 ] ]
## gap> imp2:= CommutativeDiagram( map1, map2, map3, map4, imp );
## rec( imp1 := [ ], imp2 := [ ], imp3 := [ ], imp4 := [ ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CommutativeDiagram" );
#############################################################################
##
#F CheckFixedPoints( <inside1>, <between>, <inside2> )
##
## <#GAPDoc Label="CheckFixedPoints">
## <ManSection>
## <Func Name="CheckFixedPoints" Arg='inside1, between, inside2'/>
##
## <Description>
## Let <A>inside1</A>, <A>between</A>, <A>inside2</A> be parametrized maps,
## where <A>between</A> is assumed to map each fixed point of <A>inside1</A>
## (that is, <A>inside1</A><C>[</C><M>i</M><C>] = </C><A>i</A>)
## to a fixed point of <A>inside2</A>
## (that is, <A>between</A><C>[</C><M>i</M><C>]</C> is either an integer
## that is fixed by <A>inside2</A> or a list that has nonempty intersection
## with the union of its images under <A>inside2</A>).
## <Ref Func="CheckFixedPoints"/> changes <A>between</A> and <A>inside2</A>
## by removing all those entries violate this condition.
## <P/>
## When an inconsistency is detected,
## <Ref Func="CheckFixedPoints"/> immediately returns <K>fail</K>.
## Otherwise the list of positions is returned where changes occurred.
## <P/>
## <Example><![CDATA[
## gap> subtbl:= CharacterTable( "L4(3).2_2" );;
## gap> tbl:= CharacterTable( "O7(3)" );;
## gap> fus:= InitFusion( subtbl, tbl );; fus{ [ 48, 49 ] };
## [ [ 54, 55, 56, 57 ], [ 54, 55, 56, 57 ] ]
## gap> CheckFixedPoints( ComputedPowerMaps( subtbl )[5], fus,
## > ComputedPowerMaps( tbl )[5] );
## [ 48, 49 ]
## gap> fus{ [ 48, 49 ] };
## [ [ 56, 57 ], [ 56, 57 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CheckFixedPoints" );
#############################################################################
##
#F TransferDiagram( <inside1>, <between>, <inside2>[, <improvements>] )
##
## <#GAPDoc Label="TransferDiagram">
## <ManSection>
## <Func Name="TransferDiagram"
## Arg='inside1, between, inside2[, improvements]'/>
##
## <Description>
## Let <A>inside1</A>, <A>between</A>, <A>inside2</A> be parametrized maps
## covering parametrized maps <M>m_1</M>, <M>f</M>, <M>m_2</M> with the
## property that <C>CompositionMaps</C><M>( m_2, f )</M> is equal to
## <C>CompositionMaps</C><M>( f, m_1 )</M>.
## <P/>
## <Ref Func="TransferDiagram"/> checks this consistency, and changes the
## arguments such that all possible images are removed that cannot occur in
## the parametrized maps <M>m_i</M> and <M>f</M>.
## <P/>
## So <Ref Func="TransferDiagram"/> is similar to
## <Ref Func="CommutativeDiagram"/>,
## but <A>between</A> occurs twice in each diagram checked.
## <P/>
## If a record <A>improvements</A> with fields <C>impinside1</C>,
## <C>impbetween</C>, and <C>impinside2</C> is specified,
## only those diagrams with elements of <C>impinside1</C> as preimages of
## <A>inside1</A>, elements of <C>impbetween</C> as preimages of
## <A>between</A> or elements of <C>impinside2</C> as preimages of
## <A>inside2</A> are considered.
## <P/>
## When an inconsistency is detected,
## <Ref Func="TransferDiagram"/> immediately returns <K>fail</K>.
## Otherwise a record is returned that contains three lists
## <C>impinside1</C>, <C>impbetween</C>, and <C>impinside2</C> of positions
## where the arguments were changed.
## <P/>
## <Example><![CDATA[
## gap> subtbl:= CharacterTable( "2F4(2)" );; tbl:= CharacterTable( "Ru" );;
## gap> fus:= InitFusion( subtbl, tbl );;
## gap> permchar:= Sum( Irr( tbl ){ [ 1, 5, 6 ] } );;
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