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##
#W construc.gd GAP 4 package `ctbllib' Thomas Breuer
##
#H @(#)$Id: construc.gd,v 1.14 2004/03/23 10:14:25 gap Exp $
##
#Y Copyright (C) 2002, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## 1. Character Tables of Groups of Structure $M.G.A$
## 2. Character Tables of Groups of Structure $G.S_3$
## 3. Character Tables of Groups of Structure $G.2^2$
## 4. Character Tables of Groups of Structure $2^2.G$
## 5. Brauer Tables of Extensions by $p$-regular Automorphisms
## 6. Construction Functions used in the Character Table Library
## 7. Character Tables of Coprime Central Extensions
## 8. Miscellaneous
##
Revision.( "ctbllib/gap4/construc_gd" ) :=
"@(#)$Id: construc.gd,v 1.14 2004/03/23 10:14:25 gap Exp $";
#############################################################################
#1
## The functions in this chapter deal with the construction of character
## tables from other character tables. So they fit to the functions in
## Section~"ref:Constructing Character Tables from Others" in the {\GAP}
## Reference Manual.
## But since they are used in situations that are typical for the {\GAP}
## Character Table Library, they are described here.
##
## An important ingredient of the constructions is the description of the
## action of a group automorphism on the classes by a permutation.
## In practice, these permutations are usually chosen from the group of
## table automorphisms of the character table in question
## (see~"ref:AutomorphismsOfTable" in the {\GAP} Reference Manual).
##
## Section~"Character Tables of Groups of Structure MGA" deals with
## groups of structure $M\.G\.A$, where the upwards extension $G\.A$ acts
## suitably on the central extension $M\.G$.
## Section~"Character Tables of Groups of Structure GS3" deals with
## groups that have a factor group of type $S_3$.
#T The sections~"Character Tables of Groups of Structure GV4"
#T and~"Character Tables of Groups of Structure V4G" deal with the
#T character tables of upwards and downwards extensions of a group by a
#T Klein four group.
#T Section~"Brauer Tables of Extensions by p-regular Automorphisms"
#T describes the construction of certain Brauer tables.
## Section~"Character Tables of Coprime Central Extensions" deals with
## special cases of the construction of character tables of central
## extensions from known character tables of suitable factor groups.
## Section~"Construction Functions used in the Character Table Library"
## documents the functions used to encode certain tables in the {\GAP}
## Character Table Library.
##
## Examples can be found in~\cite{Auto}.
##
#############################################################################
##
## 1. Character Tables of Groups of Structure $M.G.A$
##
#############################################################################
##
#F PossibleCharacterTablesOfTypeMGA( <tblMG>, <tblG>, <tblGA>, <aut>,
#F <identifier> )
##
## Let $H$ be a group with normal subgroups $N$ and $M$ such that
## $H/N$ is cyclic, $M \leq N$ holds,
## and such that each irreducible character of $N$
## that does not contain $M$ in its kernel induces irreducibly to $H$.
## (This is satisfied for example if $N$ has prime index in $H$
## and $M$ is a group of prime order that is central in $N$ but not in $H$.)
## Let $G = N/M$ and $A = H/N$, so $H$ has the structure $M\.G\.A$.
##
## Let <tblMG>, <tblG>, <tblGA> be the ordinary character tables of the
## groups $M\.G$, $G$, and $G\.A$, respectively,
## and <aut> the permutation of classes of <tblMG> induced by the action
## of $H$ on $M\.G$.
## Furthermore, let the class fusions from <tblMG> to <tblG> and from <tblG>
## to <tblGA> be stored on <tblMG> and <tblG>, respectively
## (see~"ref:StoreFusion" in the {\GAP} Reference Manual).
##
## `PossibleCharacterTablesOfTypeMGA' returns a list of records describing
## all possible character tables for groups $H$ that are compatible with the
## arguments.
## Note that in general there may be several possible groups $H$,
## and it may also be that ``character tables'' are constructed for which no
## group exists.
## Each of the records in the result has the following components.
## \beginitems
## `table' &
## the ordinary character table of a possible table for $H$, and
##
## `MGfusMGA' &
## the fusion map from <tblMG> into the table stored in `table'.
## \enditems
## The possible tables differ w.r.t. some power maps, and perhaps element
## orders and table automorphisms;
## in particular, the `MGfusMGA' component is the same in all records.
##
## The returned tables have the `Identifier' value <identifier>.
## The classes of these tables are sorted as follows.
## First come the classes contained in $M\.G$, sorted compatibly with the
## classes in <tblMG>, then the classes in $H \setminus M\.G$ follow,
## in the same ordering as the classes of $G\.A \setminus G$.
##
DeclareGlobalFunction( "PossibleCharacterTablesOfTypeMGA" );
#############################################################################
##
#F PossibleActionsForTypeMGA( <tblMG>, <tblG>, <tblGA> )
##
## Let the arguments be as described for `PossibleCharacterTablesOfTypeMGA'
## (see~"PossibleCharacterTablesOfTypeMGA").
## `PossibleActionsForTypeMGA' returns the set of those table automorphisms
## (see~"ref:AutomorphismsOfTable" in the {\GAP} Reference Manual)
## of <tblMG> that can be induced by the action of $H$ on $M\.G$.
##
## Information about the progress is reported if the info level of
## `InfoCharacterTable' is at least $1$
## (see~"ref:SetInfoLevel" in the {\GAP} Reference Manual).
##
DeclareGlobalFunction( "PossibleActionsForTypeMGA" );
#############################################################################
##
## 2. Character Tables of Groups of Structure $G.S_3$
##
#############################################################################
##
#F CharacterTableOfTypeGS3( <tbl>, <tbl2>, <tbl3>, <aut>, <identifier> )
#F CharacterTableOfTypeGS3( <modtbl>, <modtbl2>, <modtbl3>, <ordtbls3>,
#F <identifier> )
##
## Let $H$ be a group with a normal subgroup $G$ such that $H/G \cong S_3$,
## the symmetric group on three points,
## and let $G\.2$ and $G\.3$ be preimages of subgroups of order $2$ and $3$,
## respectively, under the natural projection onto this factor group.
##
## In the first form, let <tbl>, <tbl2>, <tbl3> be the ordinary character
## tables of the groups $G$, $G\.2$, and $G\.3$, respectively,
## and <aut> the permutation of classes of <tbl3> induced by the action
## of $H$ on $G\.3$.
## Furthermore assume that the class fusions from <tbl> to <tbl2> and <tbl3>
## are stored on <tbl>
## (see~"ref:StoreFusion" in the {\GAP} Reference Manual).
##
## In the second form, let <modtbl>, <modtbl2>, <modtbl3> be the $p$-modular
## character tables of the groups $G$, $G\.2$, and $G\.3$, respectively,
## and <ordtbls3> the ordinary character table of $H$.
##
## `CharacterTableOfTypeGS3' returns a record with the following components.
## \beginitems
## `table' &
## the ordinary or $p$-modular character table of $H$, respectively,
##
## `tbl2fustbls3' &
## the fusion map from <tbl2> into the table of $H$, and
##
## `tbl3fustbls3' &
## the fusion map from <tbl3> into the table of $H$.
## \enditems
##
## The returned table of $H$ has the `Identifier' value <identifier>.
## The classes of the table of $H$ are sorted as follows.
## First come the classes contained in $G\.3$, sorted compatibly with the
## classes in <tbl3>, then the classes in $H \setminus G\.3$ follow,
## in the same ordering as the classes of $G\.2 \setminus G$.
##
#T In fact the code is applicable in the more general case that $H/G$ is a
#T Frobenius group $F = K C$ with abelian kernel $K$ and cyclic complement
#T $C$ of prime order, see~\cite{Auto}.
#T Besides $F = S_3$, e.g., the case $F = A_4$ is interesting.
##
DeclareGlobalFunction( "CharacterTableOfTypeGS3" );
#############################################################################
##
#F PossibleActionsForTypeGS3( <tbl>, <tbl2>, <tbl3> )
##
## Let the arguments be as described for `CharacterTableOfTypeGS3'
## (see~"CharacterTableOfTypeGS3").
## `PossibleActionsForTypeGS3' returns the set of those table automorphisms
## (see~"ref:AutomorphismsOfTable" in the {\GAP} Reference Manual)
## of <tbl3> that can be induced by the action of $H$ on the classes of
## <tbl3>.
##
## Information about the progress is reported if the info level of
## `InfoCharacterTable' is at least $1$
## (see~"ref:InfoCharacterTable" in the {\GAP} Reference Manual).
##
DeclareGlobalFunction( "PossibleActionsForTypeGS3" );
#############################################################################
##
## 3. Character Tables of Groups of Structure $G.2^2$
#4
## The following functions are thought for constructing the possible
## ordinary character tables of a group of structure $G\.2^2$ from the known
## tables of the three normal subgroups of type $G\.2$.
##
#############################################################################
##
#F PossibleCharacterTablesOfTypeGV4( <tblG>, <tblsG2>, <acts>, <identifier> )
##
## Let <tblG> be the ordinary character table of a group $G$, say,
## and let <tblsG2> be a list of three character tables of groups $G\.2_1$,
## $G\.2_2$, $G\.2_3$, say, which contain $G$ as a subgroup of index $2$.
## Let <acts> be a list of three permutations describing the action of a
## group $H$, say, that contains $G\.2_1$, $G\.2_2$, $G\.2_3$ as index $2$
## subgroups which intersect in the index $4$ subgroup $G$,
## on the conjugacy classes of the corresponding tables in <tblsG2>.
## Further it is assumed that the class fusions from <tblG> into the tables
## in <tblsG2> are stored on <tblG>.
##
## `PossibleCharacterTablesOfTypeGV4' returns a list of records describing
## all possible character tables for groups $H$ that are compatible with the
## arguments; note that in general there may be several such groups.
## Each of the records has the following components.
## \beginitems
## `table' &
## the ordinary character table of a possible table for $H$, and
##
## `G2fusGV4' &
## the list of fusion maps from the tables in <tblsG2> into the table.
## \enditems
## The possible tables differ w.r.t. the irreducible characters and perhaps
## the table automorphisms;
## in particular, the `G2fusGV4' component is the same in all records.
##
## The returned tables have the `Identifier' value <identifier>.
## The classes of these tables are sorted as follows.
## First come the classes contained in $G$, sorted compatibly with the
## classes in <tblG>, then the outer classes in the tables in <tblsG2>
## follow, in the same ordering as in these tables.
##
DeclareGlobalFunction( "PossibleCharacterTablesOfTypeGV4" );
#############################################################################
##
#F PossibleActionsForTypeGV4( <tblG>, <tblsG2> )
##
## Let the arguments be as described for `PossibleCharacterTablesOfTypeGV4'
## (see~"PossibleCharacterTablesOfTypeGV4").
## `PossibleActionsForTypeGV4' returns the list of those triples
## $[ \pi_1, \pi_2, \pi_3 ]$ of permutations for which a group $H$ may exist
## that contains $G\.2_1$, $G\.2_2$, $G\.2_3$ as index $2$ subgroups
## which intersect in the index $4$ subgroup $G$.
##
## Information about the progress is reported if the level of
## `InfoCharacterTable' is at least $1$
## (see~"ref:SetInfoLevel" in the {\GAP} Reference Manual).
##
DeclareGlobalFunction( "PossibleActionsForTypeGV4" );
#############################################################################
##
## 4. Character Tables of Groups of Structure $2^2.G$
#5
## The following function is thought for constructing the possible ordinary
## character tables of a group of structure $2^2\.G$ from the known tables
## of the three factor groups modulo the normal order two subgroups in the
## Klein four group.
##
#############################################################################
##
#F PossibleCharacterTablesOfTypeV4G( <tblG>, <tbls2G>, <identifier> )
#F PossibleCharacterTablesOfTypeV4G( <tblG>, <tbl2G>, <aut>, <identifier> )
##
## Let $H$ be a group with a central subgroup $N$ of type $2^2$,
## and let $Z_1$, $Z_2$, $Z_3$ be the order $2$ subgroups of $N$.
##
## In the first form, let <tblG> be the ordinary character table of $H/N$,
## and <tbls2G> be a list of length three, the entries being the ordinary
## character tables of the groups $H/Z_i$.
## In the second form, let <tbl2G> be the ordinary character table of
## $H/Z_i$ and <aut> be a permutation;
## here it is assumed that the groups $Z_i$ are permuted under an
## automorphism $\sigma$ of order $3$ of $H$, and that $\sigma$ induces the
## permutation <aut> on the classes of <tblG>.
##
## Let the class fusions from <tblG> into the tables in <tbls2G> or <tbl2G>,
## respectively, be stored on <tblG>.
##
## `PossibleCharacterTablesOfTypeV4G' returns the list of all possible
## character tables for $H$ in this situation.
## The returned tables have the `Identifier' value <identifier>.
#T which criteria are used?
##
DeclareGlobalFunction( "PossibleCharacterTablesOfTypeV4G" );
#############################################################################
##
## 5. Brauer Tables of Extensions by $p$-regular Automorphisms
#6
## As for the construction of Brauer character tables from known tables,
## the functions `PossibleCharacterTablesOfTypeMGA'
## (see~"PossibleCharacterTablesOfTypeMGA") and `CharacterTableOfTypeGS3'
## (see~"CharacterTableOfTypeGS3") work for both ordinary and Brauer tables.
## The following function is designed specially for Brauer tables.
##
#############################################################################
##
#F BrauerTableOfExtensionBySingularAutomorphism( <modtbl>, <ordexttbl> )
##
## Let <modtbl> be a $p$-modular Brauer table of the group $G$, say,
## and <ordexttbl> the ordinary character table of an extension of $G$
## by an automorphism of order $p$.
## Further let the class fusion from the ordinary table of <modtbl> into
## <ordexttbl> be stored.
## Then `BrauerTableOfExtensionBySingularAutomorphism' returns the
## $p$-modular Brauer table of <ordexttbl>.
##
DeclareGlobalFunction( "BrauerTableOfExtensionBySingularAutomorphism" );
#############################################################################
##
## 6. Construction Functions used in the Character Table Library
#7
## The following functions are used in the {\GAP} Character Table Library,
## for encoding table constructions via the mechanism that is based on the
## attribute `ConstructionInfoCharacterTable'
## (see~"ConstructionInfoCharacterTable").
## All construction functions take as their first argument a record that
## describes the table to be constructed, and the function adds only those
## components that are not yet contained in this record.
##
#############################################################################
##
#F ConstructMGA( <tbl>, <subname>, <factname>, <plan>, <perm> )
##
## `ConstructMGA' constructs the ordinary character table <tbl> of a group
## $m\.G\.a$ where the automorphism $a$ (a group of prime order) of $m\.G$
## acts notrivially on the central subgroup $m$ of $m\.G$.
## <subname> is the name of the subgroup $m\.G$ which is a (not necessarily
## cyclic) central extension of the (not necessarily simple) group $G$,
## <factname> is the name of the factor group $G\.a$.
## Then the faithful characters of <tbl> are induced characters of $m\.G$.
##
## <plan> is a list, each entry being a list containing positions of
## characters of $m\.G$ that form an orbit under the action of $a$
## (so the induction of characters is simulated).
##
## <perm> is the permutation that must be applied to the list of characters
## that is obtained on appending the faithful characters to the
## inflated characters of the factor group.
## A nonidentity permutation occurs for example for groups of structure
## $12\.G\.2$ that are encoded via the subgroup $12\.G$ and the factor group
## $6\.G\.2$, where the faithful characters of $4\.G\.2$ shall precede those
## of $6\.G\.2$.
##
## Examples where `ConstructMGA' is used to encode library tables are the
## tables of $3\.F_{3+}\.2$ (subgroup $3\.F_{3+}$, factor group $F_{3+}\.2$)
## and $12_1\.U_4(3)\.2_2$ (subgroup $12_1\.U_4(3)$, factor group
## $6_1\.U_4(3)\.2_2$).
##
DeclareGlobalFunction( "ConstructMGA" );
DeclareSynonym( "ConstructMixed", ConstructMGA );
#############################################################################
##
#F ConstructMGAInfo( <tblmGa>, <tblmG>, <tblGa> )
##
## Let <tblmGa> be the ordinary character table of a group of structure
## $m\.G\.a$ where the factor group of prime order $a$ acts nontrivially on
## the normal subgroup of order $m$ that is central in $m\.G$,
## <tblmG> the character table of $m\.G$, and <tblGa> the character table of
## the factor group $G\.a$.
##
## `ConstructMGAInfo' returns the list that is to be stored in the library
## version of <tblmGa>: the first entry is the string `"ConstructMGA"',
## the remaining four entries are the last four arguments for the call to
## `ConstructMGA' (see~"ConstructMGA").
##
DeclareGlobalFunction( "ConstructMGAInfo" );
#############################################################################
##
#F ConstructGS3( <tbls3>, <tbl2>, <tbl3>, <ind2>, <ind3>, <ext>, <perm> )
#F ConstructGS3Info( <tbl2>, <tbl3>, <tbls3> )
##
## `ConstructGS3' constructs the irreducibles of an ordinary character table
## <tbls3> of type $G\.S_3$ from the tables with names <tbl2> and <tbl3>,
## which correspond to the groups $G\.2$ and $G\.3$, respectively.
## <ind2> is a list of numbers referring to irreducibles of <tbl2>.
## <ind3> is a list of pairs, each referring to irreducibles of <tbl3>.
## <ext> is a list of pairs, each referring to one irreducible of <tbl2>
## and one of <tbl3>.
## <perm> is a permutation that must be applied to the irreducibles
## after the construction.
##
## `ConstructGS3Info' returns a record with the components `ind2', `ind3',
## `ext', `perm', and `list', as are needed for `ConstructGS3'.
#T better return just the `list' component?
##
DeclareGlobalFunction( "ConstructGS3" );
DeclareGlobalFunction( "ConstructGS3Info" );
#############################################################################
##
#F ConstructV4G( <tbl>, <facttbl>, <aut>[, <ker>] )
##
## Let <tbl> be the character table of a group of type $2^2\.G$
## where an outer automorphism of order 3 permutes the three involutions
## in the central $2^2$.
## Let <aut> be the permutation of classes of <tbl> induced by that
## automorphism, and <facttbl> the name of the character table
## of the factor group $2\.G$.
## Then `ConstructV4G' constructs the irreducible characters of <tbl> from
## that information.
##
## The optional argument <ker> is an integer denoting the position of the
## nontrivial class of the table of $2\.G$ that lies in the kernel of the
## epimorphism onto $G$; the default for <ker> is $2$.
##
DeclareGlobalFunction( "ConstructV4G" );
#############################################################################
##
#F ConstructProj( <tbl>, <irrinfo> )
#F ConstructProjInfo( <tbl>, <kernel> )
##
## `ConstructProj' constructs the irreducible characters of record encoding
## the ordinary character table <tbl> from projective characters of tables
## of factor groups,
## which are stored in the `ProjectivesInfo' (see~"ProjectivesInfo") value
## of the smallest factor;
## the information about the name of this factor and the projectives to
## take is stored in <irrinfo>.
##
## `ConstructProjInfo' takes an ordinary character table <tbl> and a list
## <kernel> of class positions of a cyclic kernel of order dividing $12$,
## and returns a record with the components
## \beginitems
## `tbl' &
## a character table that is permutation isomorphic with <tbl>,
## and sorted such that classes that differ only by multiplication with
## elements in the classes of <kernel> are consecutive,
##
## `projectives' &
## a record being the entry for the `projectives' list of the table of
## the factor of <tbl> by <kernel>,
## describing this part of the irreducibles of <tbl>, and
##
## `info' &
## the value of <irrinfo>.
## \enditems
##
## In order to encode a library table $t$ as a ``projective table'' relative
## to another library table $f$, say, one has to do the following.
## First the factor fusion from $t$ to $f$ must be stored on the table of
## $t$, and $t$ is written to a library file.
## Then the result of `ConstructProjInfo', called for $t$ and the kernel of
## the factor fusion, is used as follows.
## The list containing `"ConstructProj"' at its first position and the
## `info' component is added as last entry of the `MOT' call for this
## library version.
## The `projectives' component is added to the `ProjectivesInfo' list of
## $f$, and a new library version of $f$ is produced (this contains the new
## projectives via an `ARC' call).
## Finally, `etc/maketbl' is called in order to store the projection for the
## factor fusion in the `ctprimar.tbl' data.
##
DeclareGlobalFunction( "ConstructProj" );
DeclareGlobalFunction( "ConstructProjInfo" );
#############################################################################
##
#F ConstructDirectProduct( <tbl>, <factors> )
#F ConstructDirectProduct( <tbl>, <factors>, <permclasses>, <permchars> )
##
## is a special case of a `construction' call for a library table <tbl>.
##
## The direct product of the tables described in the list <factors> is
## constructed, and all its components stored not yet in <tbl> are
## added to <tbl>.
##
## The `computedClassFusions' component of <tbl> is enlarged
## by the factor fusions from the direct product to the factors.
##
## If the optional arguments <permclasses>, <permchars> are given then
## classes and characters of the result are sorted accordingly.
##
## <factors> must have length at least two;
## use `ConstructPermuted' (see~"ConstructPermuted") in the case of only
## one factor.
##
DeclareGlobalFunction( "ConstructDirectProduct" );
#############################################################################
##
#F ConstructSubdirect( <tbl>, <factors>, <choice> )
##
## The library table <tbl> is completed with help of the table obtained by
## taking the direct product of the tables with names in the list <factors>,
## and then taking the table consisting of the classes in the list <choice>.
##
## Note that in general, the restriction to the classes of a normal subgroup
## is not sufficient for describing the irreducible characters of this
## normal subgroup.
##
DeclareGlobalFunction( "ConstructSubdirect" );
#############################################################################
##
#F ConstructIsoclinic( <tbl>, <factors> )
#F ConstructIsoclinic( <tbl>, <factors>, <nsg> )
##
## constructs first the direct product of library tables as given by the
## list <factors>, and then constructs the isoclinic table of the result.
##
DeclareGlobalFunction( "ConstructIsoclinic" );
#############################################################################
##
#F ConstructPermuted( <tbl>, <libnam>[, <prmclasses>, <prmchars>] )
##
## The library table <tbl> is completed with help of the library table with
## name <libnam>, whose classes and characters must be permuted by the
## permutations <prmclasses> and <prmchars>, respectively.
##
DeclareGlobalFunction( "ConstructPermuted" );
#############################################################################
##
#F ConstructFactor( <tbl>, <libnam>, <kernel> )
##
## The library table <tbl> is completed with help of the library table with
## name <libnam>, by factoring out the classes in the list <kernel>.
##
DeclareGlobalFunction( "ConstructFactor" );
#############################################################################
##
#F ConstructClifford( <tbl>, <cliffordtable> )
##
## constructs the irreducibles of the ordinary character table <tbl> from
## the Clifford matrices stored in `<tbl>.cliffordTable'.
##
DeclareGlobalFunction( "ConstructClifford" );
#############################################################################
##
## 7. Character Tables of Coprime Central Extensions
##
#############################################################################
##
#F CharacterTableOfCommonCentralExtension( <tblG>, <tblmG>, <tblnG>, <id> )
##
## Let <tblG> be the ordinary character table of a group $G$, say,
## and let <tblmG> and <tblnG> be the ordinary character tables of central
## extensions $m.G$ and $n.G$ of $G$ by cyclic groups of prime orders $m$
## and $n$, respectively, with $m \not= n$.
## We assume that the factor fusions from <tblmG> and <tblnG> to <tblG> are
## stored on the tables.
## `CharacterTableOfCommonCentralExtension' returns a record with the
## following components.
##
## \beginitems
## `tblmnG' &
## the character table $t$, say, of the corresponding central extension
## of $G$ by a cyclic group of order $m n$ that factors through $m.G$
## and $n.G$; the `Identifier' value of this table is <id>,
##
## `IsComplete' &
## `true' if the `Irr' value is stored in $t$, and `false' otherwise,
##
## `irreducibles' &
## the list of irreducibles of $t$ that are known;
## it contains the inflated characters of the factor groups $m.G$ and
## $n.G$, plus those irreducibles that were found in tensor
## products of characters of these groups.
## \enditems
##
## Note that the conjugacy classes and the power maps of $t$ are uniquely
## determined by the input data.
## Concerning the irreducible characters, we try to extract them from the
## tensor products of characters of the given factor groups by reducing
## with known irreducibles and applying the LLL algorithm
## (see~"ref:ReducedClassFunctions" and~"ref:LLL"
## in the {\GAP} Reference Manual).
##
DeclareGlobalFunction( "CharacterTableOfCommonCentralExtension" );
#############################################################################
##
#F IrreduciblesForCharacterTableOfCommonCentralExtension(
#F <tblmnG>, <factirreducibles>, <zpos>, <needed> )
##
## This function implements a heuristic for finding the missing irreducible
## characters of a character table whose table head is constructed with
## `CharacterTableOfCommonCentralExtension'
## (see~"CharacterTableOfCommonCentralExtension").
## Currently reducing tensor products and applying the LLL algorithm are
## the only ingredients.
##
DeclareGlobalFunction(
"IrreduciblesForCharacterTableOfCommonCentralExtension" );
#############################################################################
##
## 8. Miscellaneous
#8
##
#############################################################################
##
#F PossibleActionsForTypeGA( <tblG>, <tblGA> )
##
## Let <tblG> and <tblGA> be the ordinary character tables of a group $G$
## and of an extension $\tilde{G}$ of $G$ by an automorphism of order $A$,
## say.
##
## `PossibleActionsForTypeGA' returns the list of all those permutations
## that may describe the action of $\tilde{G}$ on the classes
## of <tblG>, that is, all table automorphisms of <tblG> that have order
## dividing $A$ and permute the classes of <tblG> compatibly with the fusion
## from <tblG> into <tblGA>.
##
DeclareGlobalFunction( "PossibleActionsForTypeGA" );
#T Replace the function by one that takes a perm. group and a fusion map!
#T The following two functions belong to the package for interactive
#T character table constructions;
#T but they are needed for `CharacterTableOfCommonCentralExtension'.
#############################################################################
##
#F ReducedX( <tbl>, <redresult>, <chars> )
##
## Let <tbl> be an ordinary character table, <redresult> be a result record
## returned by `Reduced' when called with first argument <tbl>, and <chars>
## be a list of characters of <tbl>.
## `ReducedX' first reduces <chars> with the `irreducibles' component of
## <redresult>; if new irreducibles are obtained this way then the
## characters in the `remainders' component of <redresult> are reduced with
## them; this process is iterated until no more irreducibles are found.
## The function returns a record with the following components.
##
## \beginitems
## `irreducibles' &
## all irreducible characters found during the process, including the
## `irreducibles' component of <redresult>,
##
## `remainders' &
## the reducible characters that are left from <chars> and the
## `remainders' component of <redresult>.
## \enditems
##
DeclareGlobalFunction( "ReducedX" );
#############################################################################
##
#F TensorAndReduce( <tbl>, <chars1>, <chars2>, <irreducibles>, <needed> )
##
## Let <tbl> be an ordinary character table, <chars1> and <chars2> be two
## lists of characters of <tbl>, <irreducibles> be a list of irreducible
## characters of <tbl>, and <needed> be a nonnegative integer.
## `TensorAndReduce' forms the tensor products of the characters in <chars1>
## with the characters in <chars2>, and reduces them with the characters in
## <irreducibles> and with all irreducible characters that are found this
## way.
## The function returns a record with the following components.
##
## \beginitems
## `irreducibles' &
## all new irreducible characters found during the process,
##
## `remainders' &
## the reducible characters that are left from the tensor products.
## \enditems
##
## When at least <needed> new irreducibles are found then the process is
## stopped immediately, without forming more tensor products.
##
## For example, <chars1> and <chars2> can be chosen as lists of irreducible
## characters with prescribed kernels such that the tensor products have a
## prescribed kernel, too.
## In this situation, <irreducibles> can be restricted to the list of those
## known irreducible characters that can be constituents of the tensor
## products, and <needed> can be chosen as the number of all missing
## irreducibles of that kind.
##
DeclareGlobalFunction( "TensorAndReduce" );
#############################################################################
##
#E
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