gap-character-tables 1r1p3-5 / usr / share / gap / pkg / ctbllib / doc / construc.tex

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%W  construc.msk          GAP 4 package `ctbllib'               Thomas Breuer
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%H  @(#)$Id: construc.msk,v 1.6 2004/03/12 09:41:42 gap Exp $
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%Y  Copyright (C) 2002,  Lehrstuhl D fuer Mathematik,   RWTH Aachen,  Germany
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\Chapter{Functions for Character Table Constructions}

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 $S_3$.
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}.




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\Section{Attributes for Character Table Constructions}

\>ConstructionInfoCharacterTable( <tbl> ) A

If this attribute is set for an ordinary character table <tbl> then
the value is a list that describes how this table was constructed.
The first entry is a string that is the identifier of the function that
was applied to the pre-table record; the remaining entries are the
arguments for that functions, except that the pre-table record must be
prepended to these arguments.




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\Section{Character Tables of Groups of Structure MGA}

\>PossibleCharacterTablesOfTypeMGA( <tblMG>, <tblG>, <tblGA>, <aut>, %
 <identifier> ) F

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 $G\.A \setminus G$.


\>PossibleActionsForTypeMGA( <tblMG>, <tblG>, <tblGA> ) F

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).




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\Section{Character Tables of Groups of Structure GS3}

\>CharacterTableOfTypeGS3( <tbl>, <tbl2>, <tbl3>, <aut>, <identifier> ) F
\>CharacterTableOfTypeGS3( <modtbl>, <modtbl2>, <modtbl3>, <ordtbls3>, %
 <identifier> ) F

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 $G\.2 \setminus G$.



\>PossibleActionsForTypeGS3( <tbl>, <tbl2>, <tbl3> ) F

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).




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%Section{Character Tables of Groups of Structure GV4}

%FileHeader[4]{construc}

%Declaration{PossibleCharacterTablesOfTypeGV4}
%Declaration{PossibleActionsForTypeGV4}


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%Section{Character Tables of Groups of Structure V4G}

%FileHeader[5]{construc}

%Declaration{PossibleCharacterTablesOfTypeV4G}


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%Section{Brauer Tables of Extensions by p-regular Automorphisms}

%FileHeader[6]{construc}

%Declaration{BrauerTableOfExtensionBySingularAutomorphism}


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\Section{Character Tables of Coprime Central Extensions}

\>CharacterTableOfCommonCentralExtension( <tblG>, <tblmG>, <tblnG>, <id> ) F

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).




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\Section{Construction Functions used in the Character Table Library}

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.



\>ConstructMGA( <tbl>, <subname>, <factname>, <plan>, <perm> ) F

`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
$6_1\.U_4(3)\.2_2$).


\>ConstructMGAInfo( <tblmGa>, <tblmG>, <tblGa> ) F

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").


\>ConstructGS3( <tbls3>, <tbl2>, <tbl3>, <ind2>, <ind3>, <ext>, <perm> ) F
\>ConstructGS3Info( <tbl2>, <tbl3>, <tbls3> ) F

`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'.


\>ConstructV4G( <tbl>, <facttbl>, <aut>[, <ker>] ) F

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 $2$.


\>ConstructProj( <tbl>, <irrinfo> ) F
\>ConstructProjInfo( <tbl>, <kernel> ) F

`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.


\>ConstructDirectProduct( <tbl>, <factors> ) F
\>ConstructDirectProduct( <tbl>, <factors>, <permclasses>, <permchars> ) F

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.


\>ConstructSubdirect( <tbl>, <factors>, <choice> ) F

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.


\>ConstructIsoclinic( <tbl>, <factors> ) F
\>ConstructIsoclinic( <tbl>, <factors>, <nsg> ) F

constructs first the direct product of library tables as given by the
list <factors>, and then constructs the isoclinic table of the result.


\>ConstructPermuted( <tbl>, <libnam>[, <prmclasses>, <prmchars>] ) F

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.


\>ConstructFactor( <tbl>, <libnam>, <kernel> ) F

The library table <tbl> is completed with help of the library table with
name <libnam>, by factoring out the classes in the list <kernel>.


%Declaration{ConstructClifford}


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%Section{Miscellaneous}

%FileHeader[8]{construc}

%Declaration{PossibleActionsForTypeGA}


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