<!-- %W ctbl.xml GAP documentation Thomas Breuer %W Alexander Hulpke %Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland %Y Copyright (C) 2002 The GAP Group
The documentation in this chapter corresponds to the declarations in the library files ctbl.gd, ctblauto.gd, ctblgrp.gd, ctblothe.gd, ctblsolv.gd.
-->
<Chapter Label="Character Tables">
<Heading>Character Tables</Heading>
<Index>tables</Index>
This chapter describes operations for <E>character tables of finite groups</E>.
<P/>
Operations for <E>characters</E> (or, more general, <E>class functions</E>) are
described in Chapter <Ref Chap="Class Functions"/>.
<P/>
For a description of the &GAP; Library of Character Tables,
see the separate manual for the &GAP; package <Package>CTblLib</Package>.
<P/>
Several examples in this chapter require the &GAP; Character Table Library
to be available.
If it is not yet loaded then we load it now.
<P/>
<Example><![CDATA[
gap> LoadPackage( "ctbllib" );
true
]]></Example>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Some Remarks about Character Theory in GAP">
<Heading>Some Remarks about Character Theory in &GAP;</Heading>
<#Include Label="[1]{ctbl}">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="History of Character Theory Stuff in GAP">
<Heading>History of Character Theory Stuff in GAP</Heading>
&GAP; provides functions for dealing with group characters since the version
&GAP; 3.1, which was released in March 1992.
The reason for adding this branch of mathematics to the topics of &GAP; was
(apart from the usefulness of character theoretic computations in general)
the insight that &GAP; provides an ideal environment for developing the
algorithms needed.
In particular, it had been decided at Lehrstuhl D für Mathematik
that the <Package>CAS</Package> system (a standalone Fortran program
together with a database of character tables, see <Cite Key="NPP84"/>)
should not be developed further and the functionality
of <Package>CAS</Package> should be made available in &GAP;.
The background was that extending <Package>CAS</Package>
(by new Fortran code) had turned out to be much less flexible
than writing analogous &GAP; library code.
<P/>
For integrating the existing character theory algorithms,
&GAP;'s memory management and long integer arithmetic were useful
as well as the list handling
–it is an important feature of character theoretic methods
that questions about groups are translated into manipulations of lists;
on the other hand, the datatype of cyclotomics
(see Chapter <Ref Var="Cyclotomics"/>)
was added to the &GAP; kernel because of the character theory algorithms.
For developing further code, also other areas of &GAP;'s library became
interesting, such as permutation groups, finite fields, and polynomials.
<P/>
The development of character theory code for &GAP; has been supported
by several DFG grants,
in particular the project
<Q>Representation Theory of Finite Groups and Finite Dimensional Algebras</Q>
(until 1991),
and the Schwerpunkt <Q>Algorithmische Zahlentheorie und Algebra</Q>
(from 1991 until 1997).
Besides that, several Diploma theses at Lehrstuhl D were concerned with
the development and/or implementation of algorithms dealing with characters
in &GAP;.
<P/>
The major contributions can be listed as follows.
<List>
<Item>
The arithmetic for the cyclotomics data type,
following <Cite Key="Zum89"/>,
was first implemented by Marco van Meegen; an alternative approach was
studied in the diploma thesis of Michael Scherner
(see <Cite Key="Scherner92"/>)
but was not efficient enough;
later Martin Schönert replaced the implementation by a better one. <!-- and since then, there have been more than 30 local improvements ... -->
</Item>
<Item>
The basic routines for characters and character tables were written
by Thomas Breuer and Götz Pfeiffer.
</Item>
<Item>
The lattice related functions, such as <Ref Func="LLL"/>,
<Ref Func="OrthogonalEmbeddings"/>,
and <Ref Func="DnLattice"/>, were implemented by Ansgar Kaup
(see <Cite Key="Kaup92"/>).
</Item>
<Item>
Functions for computing possible class fusions, possible power maps,
and table automorphisms were written by Thomas Breuer
(see <Cite Key="Bre91"/>).
</Item>
<Item>
Functions for computing possible permutation characters were written by
Thomas Breuer (see <Cite Key="Bre91"/>)
and Götz Pfeiffer (see <Cite Key="Pfe91"/>).
</Item>
<Item>
Functions for computing character tables from groups were written by
Alexander Hulpke (Dixon-Schneider algorithm,
see <Cite Key="Hulpke93"/>)
and Hans Ulrich Besche (Baum algorithm and Conlon algorithm,
see <Cite Key="Besche92"/>).
</Item>
<Item>
Functions for dealing with Clifford matrices were written by
Ute Schiffer (see <Cite Key="Schiffer94"/>).
</Item>
<Item>
Functions for monomiality questions were written by Thomas Breuer and
Erzsébet Horváth.
</Item>
</List>
<P/>
Since then, the code has been maintained and extended further by
Alexander Hulpke (code related to his implementation of the Dixon-Schneider
algorithm) and Thomas Breuer.
<P/> <!-- % D. Bauten? -->
Currently &GAP; does not provide special functionality for computing
Brauer character tables,
but there is an interface to the <Package>MOC</Package> system
(see <Cite Key="HJLP92"/>),
and the &GAP; Character Table Library contains many
known Brauer character tables.
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Creating Character Tables">
<Heading>Creating Character Tables</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Conventions for Character Tables">
<Heading>Conventions for Character Tables</Heading>
The following few conventions should be noted.
<List>
<Item>
The class of the <E>identity element</E> is expected to be the first one;
thus the degree of a character is the character value at position <M>1</M>.
</Item>
<Item>
The <E>trivial character</E> of a character table need not be the first in
the list of irreducibles.
</Item>
<Item>
Most functions that take a character table as an argument and work with
characters expect these characters as an argument, too.
For some functions, the list of irreducible characters serves as the
default, i.e, the value of the attribute
<Ref Attr="Irr" Label="for a character table"/>;
in these cases,
the <Ref Attr="Irr" Label="for a character table"/> value is
automatically computed if it was not yet known.
</Item>
<Item>
For a stored class fusion, the image table is denoted by its
<Ref Attr="Identifier" Label="for character tables"/> value;
each library table has a unique identifier by which it can be accessed
(see <Ref Sect="Accessing a Character Table from the Library"
BookName="ctbllib"/> in the manual for the
&GAP; Character Table Library),
tables constructed from groups get an identifier that is unique in the
current &GAP; session.
</Item>
</List>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="The Interface between Character Tables and Groups">
<Heading>The Interface between Character Tables and Groups</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Operators for Character Tables">
<Heading>Operators for Character Tables</Heading>
<#Include Label="[3]{ctbl}">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Attributes and Properties of Character Tables">
<Heading>Attributes and Properties for Groups and Character Tables</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Attributes and Properties only for Character Tables">
<Heading>Attributes and Properties only for Character Tables</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Normal Subgroups Represented by Lists of Class Positions">
<Heading>Normal Subgroups Represented by Lists of Class Positions</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Other Operations for Character Tables">
<Heading>Other Operations for Character Tables</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Printing Character Tables">
<Heading>Printing Character Tables</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Computing the Irreducible Characters of a Group">
<Heading>Computing the Irreducible Characters of a Group</Heading>
Several algorithms are available for computing the irreducible characters of
a finite group <M>G</M>.
The default method for arbitrary finite groups is to use the Dixon-Schneider
algorithm (see <Ref Attr="IrrDixonSchneider"/>).
For supersolvable groups,
Conlon's algorithm can be used (see ).
For abelian-by-supersolvable groups,
the Baum-Clausen algorithm for computing the irreducible representations
(see <Ref Attr="IrreducibleRepresentations"/>) can be used to compute
the irreducible characters (see <Ref Attr="IrrBaumClausen"/>).
<P/>
These functions are installed in methods for
<Ref Attr="Irr" Label="for a group"/>,
but explicitly calling one of them will <E>not</E> set the
<Ref Attr="Irr" Label="for a group"/> value of <M>G</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Representations given by modules">
<Heading>Representations Given by Modules</Heading>
This section describes functions that return certain modules of a given
group.
(Extensions by modules can be formed by the command <Ref Oper="Extensions"/>.)
The computation of irreducible characters of very large groups
may take quite some time.
On the other hand, for the expert only a few irreducible characters may be
needed,
since the other ones can be computed using character theoretic methods
such as tensoring, induction, and restriction.
Thus &GAP; provides also step-by-step routines for doing the calculations.
These routines allow one to compute some characters and to stop before all
are calculated.
Note that there is no <Q>safety net</Q>:
The routines (being somehow internal) do no error checking,
and assume the information given is correct.
<P/>
When the info level of <Ref InfoClass="InfoCharacterTable"/> if positive,
information about the progress of splitting is printed.
(The default value is zero.)
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Components of a Dixon Record">
<Heading>Components of a Dixon Record</Heading>
The <Q>Dixon record</Q> <A>D</A> returned by <Ref Func="DixonInit"/>
stores all the information that is used by the Dixon-Schneider routines
while computing the irreducible characters of a group.
Some entries, however, may be useful to know about when using the algorithm
interactively,
see <Ref Sect="An Example of Advanced Dixon-Schneider Calculations"/>.
<List>
<Mark><C>group</C></Mark>
<Item>
the group <M>G</M> of which the character table is to be computed,
</Item>
<Mark><C>conjugacyClasses</C></Mark>
<Item>
classes of <M>G</M> (all characters stored in the Dixon record
correspond to this arrangement of classes),
</Item>
<Mark><C>irreducibles</C></Mark>
<Item>
the already known irreducible characters
(given as lists of their values on the conjugacy classes),
</Item>
<Mark><C>characterTable</C></Mark>
<Item>
the <Ref Oper="CharacterTable" Label="for a group"/> value of <M>G</M>
(whose irreducible characters are not yet known),
</Item>
<Mark><C>ClassElement( <A>D</A>, <A>el</A> )</C></Mark>
<Item>
a function that returns the number of the class of <M>G</M>
that contains the element <A>el</A>.
</Item>
</List>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="An Example of Advanced Dixon-Schneider Calculations">
<Heading>An Example of Advanced Dixon-Schneider Calculations</Heading>
First, we set the appropriate info level higher.
<P/>
<Example><![CDATA[
gap> SetInfoLevel( InfoCharacterTable, 1 );
]]></Example>
<P/>
for printout of some internal results.
We now define our group, which is isomorphic to PSL<M>_4(3)</M>.
<P/>
<Example><![CDATA[
gap> g:= PrimitiveGroup(40,5);
PSL(4, 3)
gap> Size(g);
6065280
gap> d:= DixonInit( g );;
#I 29 classes
#I choosing prime 65521
gap> c:= d.characterTable;;
]]></Example>
<P/>
After the initialisation, one structure matrix is evaluated,
yielding smaller spaces and several irreducible characters.
<P/>
<Example><![CDATA[
gap> DixonSplit( d );
#I Matrix 2,Representative of Order 3,Centralizer: 5832
#I Dimensions: [ [ 1, 6 ], [ 2, 3 ], [ 4, 1 ], [ 12, 1 ] ]
2
]]></Example>
<P/>
In this case spaces of the listed dimensions are a result of the
splitting process.
The three two dimensional spaces are split successfully by combinatoric
means.
<P/>
We obtain several irreducible characters by tensor products and notify them
to the Dixon record.
<P/>
<Example><![CDATA[
gap> asp:= AntiSymmetricParts( c, d.irreducibles, 2 );;
gap> ro:= ReducedCharacters( c, d.irreducibles, asp );;
gap> Length( ro.irreducibles );
3
gap> DxIncludeIrreducibles( d, ro.irreducibles );
]]></Example>
<P/>
Finally we calculate the characters induced from all cyclic subgroups and
obtain the missing irreducibles by applying the LLL-algorithm to them.
<P/>
<Example><![CDATA[
gap> ic:= InducedCyclic( c, "all" );;
gap> ro:= ReducedCharacters( c, d.irreducibles, ic );;
gap> Length( ro.irreducibles );
0
gap> l:= LLL( c, ro.remainders );;
gap> Length( l.irreducibles );
13
]]></Example>
<P/>
The LLL returns class function objects
(see Chapter <Ref Chap="Class Functions"/>),
and the Dixon record works with character values lists.
So we convert them to a list of values
before feeding them in the machinery of the Dixon-algorithm.
<P/>
<Example><![CDATA[
gap> l.irreducibles[1];
Character( CharacterTable( PSL(4, 3) ),
[ 640, -8, -8, -8, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, E(13)^7+E(13)^8+E(13)^11, E(13)^4+E(13)^10+E(13)^12,
E(13)^2+E(13)^5+E(13)^6, E(13)+E(13)^3+E(13)^9, 0 ] )
gap> l:=List(l.irreducibles,ValuesOfClassFunction);;
gap> DxIncludeIrreducibles( d, l );
gap> Length( d.irreducibles );
29
gap> Length( d.classes );
29
]]></Example>
<P/>
It turns out we have found all irreducible characters.
As the last step, we obtain the irreducible characters and tell them to the
group.
This makes them available also to the character table.
<P/>
<Example><![CDATA[
gap> irrs:= DixontinI( d );;
#I Total:1 matrices,[ 2 ]
gap> SetIrr(g,irrs);
gap> Length(Irr(c));
29
gap> SetInfoLevel( InfoCharacterTable, 0 );
]]></Example>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Constructing Character Tables from Others">
<Heading>Constructing Character Tables from Others</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Sorted Character Tables">
<Heading>Sorted Character Tables</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Automorphisms and Equivalence of Character Tables">
<Heading>Automorphisms and Equivalence of Character Tables</Heading>
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