<!-- %W maintain.xml GAP 4 package CTblLib Thomas Breuer -->
<Chapter Label="chap:maintain">
<Heading>Maintenance Issues for the &GAP; Character Table Library</Heading>
This chapter collects examples of computations that arose
in the context of maintaining the &GAP; Character Table Library.
The sections have been added when the issues in question arose;
the dates of the additions are shown in the section titles.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:disprove">
<Heading>Disproving Possible Character Tables (November 2006)</Heading>
I do not know a necessary and sufficient criterion for checking
whether a given matrix together with a list of power maps
describes the character table of a finite group.
Examples of <E>pseudo character tables</E>
(tables which satisfy certain necessary conditions
but for which actually no group exists) have been given
in <Cite Key="Gag86"/>.
Another such example is described in
Section <Ref Subsect="subsect:pseudo"/>.
The tables in the &GAP; Character Table Library satisfy the usual tests.
However,
there are table candidates for which these tests are not good enough.
<!-- (mention that this should be run when a table is going to be added) --> <!-- (example: the candidate with nonintegral structure constants) -->
Another question would be whether a given character table
belongs to the group for which it is claimed to belong,
see Section <Ref Subsect="subsect:LyN2"/> for an example.
(This example arose from a discussion with Jack Schmidt.)
<P/>
Up to version 1.1.3 of the &GAP; Character Table Library,
the table with identifier <C>"P41/G1/L1/V4/ext2"</C> was not correct.
The problem occurs already in the microfiches
that are attached to <Cite Key="HP89"/>.
<P/>
In the following, we show that this table is not the character table
of a finite group,
using the &GAP; library of perfect groups.
Currently we do not know how to prove this inconsistency
alone from the table.
<P/>
We start with the construction of the inconsistent table;
apart from a little editing,
the following input equals the data formerly stored
in the file <F>data/ctoholpl.tbl</F> of the &GAP; Character Table Library.
The table satisfies the orthogonality relations,
the structure constants are nonnegative integers,
and symmetrizations of the irreducibles decompose
into the irreducibles, with nonnegative integral coefficients.
The &GAP; Library of Perfect Groups contains representatives of the
four isomorphism types of perfect groups of order <M>|G| = 64\,512</M>.
<P/>
<Example><![CDATA[
gap> n:= Size( tbl );
64512
gap> NumberPerfectGroups( n );
4
gap> grps:= List( [ 1 .. 4 ], i -> PerfectGroup( IsPermGroup, n, i ) );
[ L2(8) 2^6 E 2^1, L2(8) N 2^6 E 2^1 I, L2(8) N 2^6 E 2^1 II,
L2(8) N 2^6 E 2^1 III ]
]]></Example>
<P/>
If we believe that the classification of perfect groups of order <M>|G|</M>
is correct then all we have to do is to show that none of the
character tables of these four groups is equivalent to the given table.
Let us look closer at the tables in question.
Each character table of a perfect group of order <M>64\,512</M>
has exactly one irreducible character of degree <M>63</M> that takes exactly
the values <M>-1</M>, <M>0</M>, <M>7</M>, and <M>63</M>;
moreover, the value <M>7</M> occurs in exactly two classes.
<P/>
<Example><![CDATA[
gap> testchars:= List( tbls,
> t -> Filtered( Irr( t ),
> x -> x[1] = 63 and Set( x ) = [ -1, 0, 7, 63 ] ) );;
gap> List( testchars, Length );
[ 1, 1, 1, 1 ]
gap> List( testchars, l -> Number( l[1], x -> x = 7 ) );
[ 2, 2, 2, 2 ]
]]></Example>
<P/>
(Another way to state this is that in each of the four tables <M>t</M> in
question,
there are ten preimage classes of the involution class in the simple
factor group <M>L_2(8)</M>,
there are eight preimage classes of this class in the factor group
<M>2^6.L_2(8)</M>,
and that the unique class in which an irreducible degree <M>63</M> character
of this factor group takes the value <M>7</M> splits in <M>t</M>.)
<P/>
In the erroneous table, however,
there is only one class with the value <M>7</M> in this character.
<P/>
<Example><![CDATA[
gap> testchars:= List( [ tbl ],
> t -> Filtered( Irr( t ),
> x -> x[1] = 63 and Set( x ) = [ -1, 0, 7, 63 ] ) );;
gap> List( testchars, Length );
[ 1 ]
gap> List( testchars, l -> Number( l[1], x -> x = 7 ) );
[ 1 ]
]]></Example>
<P/>
This property can be checked easily for the displayed table stored
in fiche <M>2</M>, row <M>4</M>, column <M>7</M> of <Cite Key="HP89"/>,
with the name <C>6L1<>Z^7<>L2(8); V4; MOD 2</C>,
and it turns out that this table is not correct.
<P/>
Note that these microfiches contain <E>two</E> tables of order <M>64\,512</M>,
and there were <E>three</E> tables of groups of that order
in the &GAP; Character Table Library
that contain <C>origin: Hanrath library</C> in their
<Ref Func="InfoText" BookName="ref"/>
value.
Besides the incorrect table, these library tables are
the character tables of the groups
<C>PerfectGroup( 64512, 1 )</C> and <C>PerfectGroup( 64512, 3 )</C>,
respectively.
(The matrices of irreducible characters of these tables are equivalent.)
Since version 1.2 of the &GAP; Character Table Library,
the character table with the
<Ref Func="Identifier" BookName="ref"/>
value
<C>"P41/G1/L1/V4/ext2"</C> corresponds to the group
<C>PerfectGroup( 64512, 4 )</C>.
The choice of this group was somewhat arbitrary since the vector system
<C>V4</C> seems to be not defined in <Cite Key="HP89"/>;
anyhow, this group and the remaining perfect group,
<C>PerfectGroup( 64512, 2 )</C>,
have equivalent matrices of irreducibles.
<!-- % Let us suppose that we are not convinced yet, % perhaps because it might be that a perfect group of order <M>64\,512</M> % has been overlooked up to now. % Then we consider the factor group <M>F</M> of <M>G</M> modulo its centre. % % <Example><![CDATA[ % gap> cen:= ClassPositionsOfCentre( tbl ); % [ 1, 4 ] % gap> facttbl:= tbl / cen; % CharacterTable( "P41/G1/L1/V4/ext2/[ 1, 4 ]" ) % ]]></Example> % % The group <M>F</M> is a perfect group of order <M>32\,256</M>. % According to the &GAP; Library of Perfect Groups, % there are exactly two such groups, up to isomorphism, % and <M>F</M> must be isomorphic to the second of these groups. % % <Example><![CDATA[ % gap> factgrps:= List( [ 1 .. 2 ], i -> PerfectGroup( IsPermGroup, n/2, i ) ); % [ L2(8) 2^6, L2(8) N 2^6 ] % gap> facttbls:= List( factgrps, CharacterTable );; % gap> List( facttbls, x -> IsRecord( % > TransformingPermutationsCharacterTables( x, facttbl ) ) ); % [ false, true ] % ]]></Example> % % (In fact the situation is a bit more subtle: % The matrices of irreducibles of the two perfect groups of order <M>32\,256</M> % are equivalent, % the two character tables differ just by their second power maps and, % as a consequence, by their element orders. % However, the second power map of the given table of <M>G</M> % is uniquely determined by the matrix of irreducibles of this table.) % % No!! % % -> what about element orders? % % <Example><![CDATA[ % gap> IsRecord( TransformingPermutations( % > Irr( facttbls[1] ), Irr( facttbls[2] ) ) ); % true % gap> IsRecord( TransformingPermutationsCharacterTables( % > facttbls[1], facttbls[2] ) ); % false % % ]]></Example> % % % ... % % (And the power maps are uniquely det. by the matrix.) % % (and not equiv. to the other! % note that one is split, the other nonsplit, % and the matrices of irreds are equivalent) % % So if we believe that there is no other perfect group of order <M>32\,256</M> % that has the same character table % then <M>G</M> is a central extension of this group by a group of order two. % % We compute the possible central extensions of a maximal subgroup % of the structure <M>2^6.2^3:7</M>, % and show that none of them admits an embedding into the table. % % (Note that the extensions can be computed for solvable groups, % that's why we switch to a subgroup.) % % <Example><![CDATA[ % gap> sort; % CharacterTable( "P41/G1/L1/V4/ext2" ) % gap> g:= PerfectGroup( IsPermGroup, 32256, 2 );; % gap> nsg:= NormalSubgroups( g ); % ... % gap> epi:= NaturalHomomorphismByNormalSubgroup( g, nsg[2] );; % gap> img:= Image( epi );; % gap> mx:= MaximalSubgroupClassReps( img ); % [ Group([ (3,4,8,6,7,5,9), (1,2)(4,9)(5,8)(6,7) ]), Group([ (1,2,3)(4,7,8)(5,6,9), (1,4,9,2,7,5,3,8,6), (2,3)(4,6)(5,7)(8,9) ]), Group([ (2,3)(4,6)(5,7)(8,9), (2,4)(3,6)(5,9)(7,8), (2,5)(3,7)(4,9)(6,8), % (3,4,8,6,7,5,9) ]) ] % gap> List( mx, Size ); % [ 14, 18, 56 ] % gap> pre:= PreImages( epi, mx[3] );; % gap> iso:= IsomorphismPcGroup( pre );; % gap> G:= Image( iso );; % gap> mats:= List( Pcgs( G ), x -> IdentityMat( 1, GF(2) ) );; % gap> M:= GModuleByMats( mats, GF(2) );; % gap> ext:= Extensions( G, M );; % gap> Length( ext ); % 8 % gap> List( ext, Size ); % [ 7168, 7168, 7168, 7168, 7168, 7168, 7168, 7168 ] % gap> tbls:= List( ext, CharacterTable );; % gap> List( tbls, t -> Length( PossibleClassFusions( t, sort ) ) ); % [ 0, 0, 0, 0, 0, 0, 0, 0 ] % ]]></Example> % % % now: how to replace the wrong table by the correct one! % -> to which of the two? % -> note that V1/ext2 and V2/ext2 belong to the first and third group, % they have equiv. matrices; % V4/ext2 is wrong, and there are the second and fourth group, % again with equiv. matrices but different power maps! % (so add two tables? with which names?) % % <Example><![CDATA[ % ]]></Example> %
-->
</Subsection>
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<Subsection Label="subsect:errorE62">
<Heading>An Error in the Character Table of <M>E_6(2)</M> (March 2016)</Heading>
In March 2016, Bill Unger computed the character table of the simple group
<M>E_6(2)</M> with Magma (see <Cite Key="CP96"/>)
and compared it with the table that was contained in the
&GAP; Character Table Library since 2000.
It turned out that the two tables did not coincide.
<P/>
The differences concern irrational character values on classes of element
order <M>91</M> and power map values on these classes.
(The character values and power maps fit to each other in both tables;
thus it may be that the assumption of a wrong power has implied the wrong
character values, or vice versa.)
Specifically, the <M>11</M>th power map in the &GAP; table
fixed all elements of order <M>91</M>.
Using the smallest matrix representation of <M>E_6(2)</M> over the field with
two elements, one can easily find an element <M>g</M> of order <M>91</M>,
and show that the characteristic polynomials of <M>g</M> and <M>g^{11}</M>
differ.
Hence these two elements cannot be conjugate in <M>E_6(2)</M>.
In other words, the &GAP; table was wrong.
<P/>
<Example><![CDATA[
gap> g:= AtlasGroup( "E6(2)" );;
gap> repeat x:= PseudoRandom( g ); until Order( x ) = 91;
gap> CharacteristicPolynomial( x ) = CharacteristicPolynomial( x^11 );
false
]]></Example>
<P/>
The wrong &GAP; table has been corrected in version 1.3.0 of the
&GAP; Character Table Library.
<!-- The wrong table involved irrationalities E(91)^6+E(91)^20+E(91)^31+E(91)^38+E(91)^48+E(91)^54+E(91)^66+E(91)^68 +E(91)^69+E(91)^73+E(91)^75+E(91)^89, and the new one contains irrationalities E(91)^17+E(91)^27+E(91)^34+E(91)^45+E(91)^54+E(91)^59+E(91)^68+E(91)^75 +E(91)^83+E(91)^87+E(91)^89+E(91)^90.
These values define different subfields of CF(91). -->
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:error2F422">
<Heading>An Error in a Power Map of the Character Table of <M>2.F_4(2).2</M> (November 2015)</Heading>
As a part of the computations for <Cite Key="BMO17"/>,
the character table of the group <M>2.F_4(2).2</M> was computed
automatically from a representation of the group,
using Magma (see <Cite Key="CP96"/>).
It turned out that the <M>2</M>-nd power map that had been stored on the
library character table of <M>2.F_4(2).2</M> had been wrong.
<P/>
In fact, this was the one and only case of a power map for an &ATLAS; group
which was not determined by the character table,
and the <Ref Attr="InfoText" BookName="ref"/> value of the character table
had mentioned the two alternatives.
<P/>
Note that the ambiguity is not present in the table of the factor group
<M>F_4(2).2</M>, and only four faithful irreducible characters of
<M>2.F_4(2).2</M> distinguish the four relevant conjugacy classes.
I had not found a suitable subgroup of <M>2.F_4(2).2</M> whose
character table could be used to decide the question which of the
two alternatives is the correct one.
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:LyN2">
<Heading>A Character Table with a Wrong Name (May 2017)</Heading>
(This example is much older.)
<P/>
The character table that is shown in <Cite Key="Ost86" Where="p. 126 f."/>
is claimed to be the table of a Sylow <M>2</M> subgroup <M>P</M> of the
sporadic simple Lyons group <M>Ly</M>.
This table had been contained in the character table library of the
<Package>CAS</Package> system (see <Cite Key="NPP84"/>),
which was one of the predecessors of &GAP;.
<P/>
It is easy to see that no subgroup of <M>Ly</M>
can have this character table.
Namely,
the group of that table contains elements of order eight
with centralizer order <M>2^6</M>,
and this does not occur in <M>Ly</M>.
The table of <M>P</M> has been computed in <Cite Key="Bre91"/>
with character theoretic methods.
Nowadays it would be no problem to take a permutation representation of
<M>Ly</M>, to compute its Sylow <M>2</M> subgroup, and use this group
to compute its character table.
However, the task is even easier if we assume that <M>Ly</M> has a subgroup
of the structure <M>3.McL.2</M>.
This subgroup is of odd index, hence it contains a conjugate of <M>P</M>.
Clearly the Sylow <M>2</M> subgroups in the factor group <M>McL.2</M>
are isomorphic with <M>P</M>.
Thus we can start with a rather small permutation representation.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:spacegroupfactors">
<Heading>Some finite factor groups of perfect space groups (February 2014)</Heading>
If one wants to find a group to which a given character table from the
&GAP; Character Table Library belongs,
one can try the function
<Ref Func="GroupInfoForCharacterTable" BookName="ctbllib"/>.
For a long time,
this was not successful in the case of <M>16</M> character tables
that had been computed by W. Hanrath (see Section
<Q>Ordinary and Brauer Tables in the &GAP; Character Table Library</Q>
in the &CTblLib; manual).
<P/>
Using the information from <Cite Key="HP89"/>,
it is straightforward to construct
such groups as factor groups of infinite groups.
Since version 1.3.0 of the &CTblLib; package,
calling
<Ref Func="GroupInfoForCharacterTable" BookName="ctbllib"/>
for the <M>16</M> library tables
in question yields nonempty lists and thus allows one to access the
results of these constructions,
via the function <C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.
This is an undocumented auxiliary function that becomes available
automatically when
<Ref Func="GroupInfoForCharacterTable" BookName="ctbllib"/>
has been called for the first time.
Below we list the <M>16</M> group constructions.
In each case, an epimorphism from the space group in question is defined by
mapping the generators returned by by the function
<C>generatorsOfPerfectSpaceGroup</C> defined below to the generators
stored in the attribute
<Ref Func="GeneratorsOfGroup" BookName="ref"/>
of the group returned by
<C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:constructspacegroups">
<Heading>Constructing the space groups in question</Heading>
In <Cite Key="HP89"/>,
a space group <M>S</M> is described as a subgroup
<M>\{ M(g, t); g \in P, t \in T \}</M> of GL<M>(d+1, &ZZ;)</M>,
where
the <E>point group</E> <M>P</M> of <M>S</M> is a finite subgroup of
GL<M>(d, &ZZ;)</M>,
the <E>translation lattice</E> <M>T</M> of <M>S</M> is a sublattice of
<M>&ZZ;^d</M>,
and the <E>vector system</E> <M>V</M> of <M>S</M> is a map from <M>P</M> to
<M>&ZZ;^d</M>.
Note that <M>V</M> maps the identity matrix <M>I \in</M> GL<M>(d, &ZZ;)</M>
to the zero vector,
and <M>M(T):= \{ M(I, t); t \in T \}</M>
is a normal subgroup of <M>S</M> that is isomorphic with <M>T</M>.
More generally, <M>M(n T)</M> is a normal subgroup of <M>S</M>,
for any positive integer <M>n</M>.
<P/>
Specifically,
<M>P</M> is given by generators <M>g_1, g_2, \ldots, g_k</M>,
<M>T</M> is given by a <M>&ZZ;</M>-basis
<M>B = \{ b_1, b_2, \ldots, b_d \}</M> of <M>T</M>,
and <M>V</M> is given by the vectors <M>V(g_1), V(g_2), \ldots, V(g_k)</M>.
<P/>
In the examples below, the matrix representation of <M>P</M> is irreducible,
so we need just the following <M>k+1</M> elements to generate <M>S</M>:
These generators are returned by the function
<C>generatorsOfPerfectSpaceGroup</C>,
when the inputs are <M>[ g_1, g_2, \ldots, g_k ]</M>,
<M>[ V(g_1), V(g_2), \ldots, V(g_k) ]</M>, and <M>b_1</M>.
<P/>
<Example><![CDATA[
gap> generatorsOfPerfectSpaceGroup:= function( Pgens, V, t )
> local d, result, i, m;
> d:= Length( Pgens[1] );
> result:= [];
> for i in [ 1 .. Length( Pgens ) ] do
> m:= IdentityMat( d+1 );
> m{ [ 1 .. d ] }{ [ 1 .. d ] }:= Pgens[i];
> m[ d+1 ]{ [ 1 .. d ] }:= V[i];
> result[i]:= m;
> od;
> m:= IdentityMat( d+1 );
> m[ d+1 ]{ [ 1 .. d ] }:= t;
> Add( result, m );
> return result;
> end;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:constructfactors">
<Heading>Constructing the factor groups in question</Heading>
The space group <M>S</M> acts on <M>&ZZ;^d</M>,
via <M>v \cdot M(g, t) = v g + V(g) + t</M>.
A (not necessarily faithful) representation of <M>S/M(n T)</M>
can be obtained from the corresponding action of <M>S</M> on
<M>&ZZ;^d/(n &ZZ;^d)</M>,
that is, by reducing the vectors modulo <M>n</M>.
For the &GAP; computations, we work instead with vectors of length <M>d+1</M>,
extending each vector in <M>&ZZ;^d</M> by <M>1</M> in the last position,
and acting on these vectors by right multiplicaton with elements of <M>S</M>.
Multiplication followed by reduction modulo <M>n</M> is implemented by
the action function returned by <C>multiplicationModulo</C>
when this is called with argument <M>n</M>.
<P/>
<Example><![CDATA[
gap> multiplicationModulo:= n -> function( v, g )
> return List( v * g, x -> x mod n ); end;;
]]></Example>
<P/>
In some of the examples,
the representation of <M>P</M> given in <Cite Key="HP89"/>
is the action on the factor
of a permutation module modulo its trivial submodule.
For that, we provide the function <C>deletedPermutationMat</C>,
cf. <Cite Key="HP89" Where="p. 269"/>.
<P/>
<Example><![CDATA[
gap> deletedPermutationMat:= function( pi, n )
> local mat, j, i;
> mat:= PermutationMat( pi, n );
> mat:= mat{ [ 1 .. n-1 ] }{ [ 1 .. n-1 ] };
> j:= n ^ pi;
> if j <> n then
> for i in [ 1 .. n-1 ] do
> mat[i][j]:= -1;
> od;
> fi;
> return mat;
> end;;
]]></Example>
<P/>
After constructing permutation generators for the example groups,
we verify that the groups fit to the character tables from the
&GAP; Character Table Library and to the permutation generators
stored for the construction of the group via
<C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.
<P/>
<!--
(In earlier versions of &GAP;, a call of the function <Ref Func="SmallerDegreePermutationRepresentation" BookName="ref"/> was sufficient for computing a reasonably small permutation representation of each of the example groups. Meanwhile, this function has become more sophisticated, with the effect that it requires much more space in some of our cases. Therefore, we reduce the number of points by hand.)
<Example><![CDATA[ gap> verifyFactorGroup:= function( gens, id ) > local g, s, sgens, act, sm, stored, hom; > # sm:= SmallerDegreePermutationRepresentation( Group( gens ) ); > g:= Group( gens ); > s:= NormalClosure( g, Subgroup( g, [ gens[ Length( gens ) ] ] ) ); > sgens:= MinimalGeneratingSet( s ); > s:= Subgroup( s, sgens{ [ 2 .. Length( sgens ) ] } ); > act:= Action( g, RightTransversal( g, s ), OnRight ); > if Size( act ) <> Size( g ) then > Error( "wrong group order!" ); > fi; > gens:= GeneratorsOfGroup( act ); > sm:= SmallerDegreePermutationRepresentation( act ); > gens:= List( gens, x -> x^sm ); > act:= Images( sm ); > if not IsRecord( TransformingPermutationsCharacterTables( > CharacterTable( act ), > CharacterTable( id ) ) ) then > return "wrong character table"; > fi; > GroupInfoForCharacterTable( id ); > stored:= CTblLib.FactorGroupOfPerfectSpaceGroup( id ); > hom:= GroupHomomorphismByImages( stored, act, > GeneratorsOfGroup( stored ), gens ); > if hom = fail or not IsBijective( hom ) then > return "wrong group"; > fi; > return true; > end;; ]]></Example>
-->
<P/>
<Example><![CDATA[
gap> verifyFactorGroup:= function( gens, id )
> local sm, act, stored, hom;
> sm:= SmallerDegreePermutationRepresentation( Group( gens ) );
> gens:= List( gens, x -> x^sm );
> act:= Images( sm );
> if not IsRecord( TransformingPermutationsCharacterTables(
> CharacterTable( act ),
> CharacterTable( id ) ) ) then
> return "wrong character table";
> fi;
> GroupInfoForCharacterTable( id );
> stored:= CTblLib.FactorGroupOfPerfectSpaceGroup( id );
> hom:= GroupHomomorphismByImages( stored, act,
> GeneratorsOfGroup( stored ), gens );
> if hom = fail or not IsBijective( hom ) then
> return "wrong group";
> fi;
> return true;
> end;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-A5">
<Heading>Examples with point group <M>A_5</M></Heading>
There are two examples with <M>d = 5</M>.
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 272"/>).
The library character table with identifier <C>"P1/G2/L1/V2/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(4 L)</M>, so we compute the action on an orbit modulo <M>8</M>.
<P/>
<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P1/G2/L1/V2/ext4" );
true
]]></Example>
<P/>
In the second example,
the translation lattice is the sublattice <M>2 L_2</M> of <M>&ZZ;^d</M>
where <M>L_2</M> has the following basis.
For the sake of simplicity, we rewrite the action of the point group
to one on <M>L_2</M>, and we adjust also the vector system.
<P/>
<Example><![CDATA[
gap> B:= Basis( Rationals^Length( bas ), bas );;
gap> abas:= List( bas, x -> Coefficients( B, x * a ) );;
gap> bbas:= List( bas, x -> Coefficients( B, x * b ) );;
gap> vbas:= List( v, x -> Coefficients( B, x ) );
[ [ 3/2, 1, 2, 3/2, -1 ], [ 0, 0, 0, 0, 0 ] ]
]]></Example>
<P/>
In order to work with integral matrices (which is necessary because
<C>multiplicationModulo</C> uses &GAP;'s mod operator),
we double both the vector system and the translation lattice.
The library character table with identifier <C>"P1/G2/L2/V2/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(8 L_2)</M>;
since we have doubled the lattice,
we compute the action on an orbit modulo <M>16</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-L32">
<Heading>Examples with point group <M>L_3(2)</M></Heading>
There are three examples with <M>d = 6</M>
and one example with <M>d = 8</M>.
The generators of the point group for the first three examples are as follows
(see <Cite Key="HP89" Where="p. 290"/>).
The first vector system is the trivial vector system <M>V_1</M>
(that is, the space group <M>S</M> is a split extension of the point group
and the translation lattice),
and the translation lattice is the full lattice <M>L_1 = &ZZ;^d</M>.
<P/>
The library character table with identifier <C>"P11/G1/L1/V1/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(4 L_1)</M>, so we compute the action on an orbit modulo <M>4</M>.
The second vector system is <M>V_2</M>,
and the translation lattice is <M>2 L_1</M>.
<P/>
The library character table with identifier <C>"P11/G1/L1/V2/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(8 L_1)</M>, so we compute the action on an orbit modulo <M>8</M>.
The third vector system is <M>V_3</M>,
and the translation lattice is <M>2 L_1</M>.
<P/>
The library character table with identifier <C>"P11/G1/L1/V3/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(8 L_1)</M>, so we compute the action on an orbit modulo <M>8</M>.
The vector system is the trivial vector system <M>V_1</M>,
and the translation lattice is the full lattice <M>L_1 = &ZZ;^d</M>.
<P/>
The library character table with identifier <C>"P11/G4/L1/V1/ext3"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L_1)</M>, so we compute the action on an orbit modulo <M>3</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-sl27">
<Heading>Example with point group SL<M>_2(7)</M></Heading>
There is one example with <M>d = 8</M>.
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 295"/>).
The vector system is the trivial vector system <M>V_1</M>,
and the translation lattice is the sublattice <M>L_2</M> of <M>&ZZ;^d</M>
that has the following basis,
which is called <M>B(2,8)</M> in <Cite Key="HP89" Where="p. 269"/>.
For the sake of simplicity, we rewrite the action to one on <M>L_2</M>.
<P/>
<Example><![CDATA[
gap> B:= Basis( Rationals^Length( bas ), bas );;
gap> abas:= List( bas, x -> Coefficients( B, x * a ) );;
gap> bbas:= List( bas, x -> Coefficients( B, x * b ) );;
]]></Example>
<P/>
The library character table with identifier <C>"P12/G1/L2/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_2)</M>.
The action on an orbit modulo <M>2</M> is not faithful,
its kernel contains the centre of SL<M>(2,7)</M>.
We can compute a faithful representation by acting on pairs:
One entry is the usual vector and the other entry carries the action
of the point group.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-23L32">
<Heading>Example with point group <M>2^3.L_3(2)</M></Heading>
There is one example with <M>d = 7</M>.
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 297"/>).
The vector system is the trivial vector system <M>V_1</M>,
and the translation lattice is the sublattice <M>L_2</M> of <M>&ZZ;^d</M>
that has the following basis,
which is called <M>B(2,7)</M> in <Cite Key="HP89" Where="p. 269"/>.
For the sake of simplicity, we rewrite the action to one on <M>L_2</M>.
<P/>
<Example><![CDATA[
gap> B:= Basis( Rationals^Length( bas ), bas );;
gap> abas:= List( bas, x -> Coefficients( B, x * a ) );;
gap> bbas:= List( bas, x -> Coefficients( B, x * b ) );;
gap> cbas:= List( bas, x -> Coefficients( B, x * c ) );;
]]></Example>
<P/>
The library character table with identifier <C>"P13/G1/L2/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_2)</M>, so we compute the action on an orbit modulo <M>2</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-A6">
<Heading>Examples with point group <M>A_6</M></Heading>
There are two examples with <M>d = 10</M>.
In both cases, the generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 307"/>).
In both examples, the vector system is the trivial vector system <M>V_1</M>,
and the translation lattices are the lattices <M>L_2</M> and <M>L_5</M>,
respectively, which have the following bases.
For the sake of simplicity, we rewrite the action to actions on <M>L_2</M>
and <M>L_5</M>, respectively.
<P/>
<Example><![CDATA[
gap> B2:= Basis( Rationals^Length( bas2 ), bas2 );;
gap> bbas2:= List( bas2, x -> Coefficients( B2, x * b ) );;
gap> cbas2:= List( bas2, x -> Coefficients( B2, x * c ) );;
gap> B5:= Basis( Rationals^Length( bas5 ), bas5 );;
gap> bbas5:= List( bas5, x -> Coefficients( B5, x * b ) );;
gap> cbas5:= List( bas5, x -> Coefficients( B5, x * c ) );;
]]></Example>
<P/>
The library character table with identifier <C>"P21/G3/L2/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_2)</M>, so we compute the action on an orbit modulo <M>2</M>.
The library character table with identifier <C>"P21/G3/L5/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_5)</M>, so we compute the action on an orbit modulo <M>2</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-L28">
<Heading>Examples with point group <M>L_2(8)</M></Heading>
There are two examples with <M>d = 7</M>.
In both cases, the generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 327"/>).
The library character table with identifier <C>"P41/G1/L1/V3/ext3"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L)</M>, so we compute the action on an orbit modulo <M>9</M>.
<P/>
The orbits in this action are quite long.
we choose a seed vector from the fixed space of an element of order <M>7</M>.
The library character table with identifier <C>"P41/G1/L1/V4/ext3"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(6 L)</M>, so we compute the action on an orbit modulo <M>18</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-M11">
<Heading>Example with point group <M>M_{11}</M></Heading>
There is one example with <M>d = 10</M>.
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 334"/>).
The library character table with identifier <C>"P48/G1/L1/V2/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L)</M>, so we compute the action on an orbit modulo <M>4</M>.
<P/>
<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 4 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P48/G1/L1/V2/ext2" );
true
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-U33">
<Heading>Example with point group <M>U_3(3)</M></Heading>
There is one example with <M>d = 7</M>.
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 335"/>).
The library character table with identifier <C>"P49/G1/L1/V2/ext3"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L)</M>, so we compute the action on an orbit modulo <M>9</M>.
<P/>
<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 9 );;
]]></Example>
<P/>
The orbits in this action are quite long.
we choose a seed vector from the fixed space of an element of order <M>12</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-U42">
<Heading>Examples with point group <M>U_4(2)</M></Heading>
There are two examples with <M>d = 6</M>.
In both cases, the generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 336"/>).
The library character table with identifier <C>"P50/G1/L1/V1/ext3"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L_1)</M>, so we compute the action on an orbit modulo <M>3</M>.
<P/>
<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 3 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P50/G1/L1/V1/ext3" );
true
]]></Example>
<P/>
The library character table with identifier <C>"P50/G1/L1/V1/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(4 L_1)</M>, so we compute the action on an orbit modulo <M>4</M>.
<P/>
<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 4 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P50/G1/L1/V1/ext4" );
true
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:A-remark-on-one-of-the-example-groups">
<Heading>A remark on one of the example groups</Heading>
The (perfect) character table with identifier <C>"P1/G2/L2/V2/ext4"</C>
has the property that its character degrees are exactly
the divisors of <M>60</M>.
There are nilpotent groups with the same set of character degrees,
for example the direct product of four extraspecial groups of the orders
<M>2^3</M>, <M>2^3</M>, <M>3^3</M>, and <M>5^3</M>, respectively.
This phenomenon has been described in <Cite Key="NR14"/>.
The term <Q>generality problem</Q> is used for problems concerning
consistent choices of conjugacy classes of Brauer tables for the same
group, in different characteristics.
The definition and some examples are given
in <Cite Key="JLPW95" Where="p. x"/>.
<P/>
Section <Ref Subsect="subsect:generalityproblems_list"/>
shows how to detect generality problems
and lists the known generality problems,
and Section <Ref Subsect="subsect:generality_J3"/> gives an example
that actually arose.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:generalityproblems_list">
<Heading>Listing possible generality problems</Heading>
We use the following idea for finding character tables which
may involve generality problems.
(The functions shown in this section are based on &GAP; 3 code
that was originally written by Jürgen Müller.)
<P/>
If the <M>p</M>-modular Brauer table <M>mtbl</M>, say,
of a group contributes to a generality problem
then some choice of conjugacy classes is necessary
in order to write down this table,
in the sense that some symmetry of the corresponding ordinary table
<M>tbl</M>, say, is broken in <M>mtbl</M>.
This situation can be detected as follows.
We assume that the class fusion from <M>mtbl</M> to <M>tbl</M>
has been fixed.
All possible class fusions are obtained as the orbit of this
class fusion under the actions of table automorphisms of <M>tbl</M>,
via mapping the images of the class fusion
(with the function <Ref Func="OnTuples" BookName="ref"/>),
and of the table automorphisms of <M>mtbl</M>,
via permuting the preimages.
The case of broken symmetries occurs if and only if this orbit
splits into several orbits when only the action of the
table automorphisms of <M>mtbl</M> is considered.
Equivalently, symmetries are broken if and only if the orbit under
table automorphisms of <M>mtbl</M> is not closed under the action of
table automorphisms of <M>tbl</M>.
<!-- It is sufficient to test generators of 'taut' because 'orb' is invariant under the action of 'taut'
if and only if everey generator leaves 'orb' invariant. -->
<E>Remark:</E> (Thanks to Klaus Lux for discussions on this topic.)
<List>
<Item>
It may happen that some symmetry <M>\sigma_m</M> of a Brauer table
does not belong to a symmetry <M>\sigma_o</M> of the corresponding
ordinary table,
in the sense that permuting the preimage classes of a fusion <M>f</M>
between the two tables with <M>\sigma_m</M>
and permuting the image classes with <M>\sigma_o</M> yields <M>f</M>.
<P/>
For example, consider the group <M>G = 2.A_6.2_1</M>,
the double cover of the symmetric group <M>S_6</M> on six points.
The <M>2</M>-modular Brauer table of <M>G</M>,
which is essentially equal to that of <M>S_6</M>,
has a table automorphism group order two,
and the nonidentity element in it swaps the two classes
of element order three.
The automorphism group of the ordinary character table of <M>G</M>,
however, fixes the two classes of element order three;
note that exactly one of these classes possesses square roots in the
<Q>outer half</Q> <M>G \setminus G'.
<P/>
Thus it is not sufficient to compare the orbit of the fixed class fusion
under the automorphisms of the ordinary table with the orbit of the
same fusion under the automorphisms of the Brauer table.
</Item>
</List>
<Example><![CDATA[
gap> t:= CharacterTable( "2.A6.2_1" );;
gap> m:= t mod 2;;
gap> GetFusionMap( m, t );
[ 1, 4, 6, 9 ]
gap> AutomorphismsOfTable( t );
Group([ (16,17), (14,15), (14,15)(16,17) ])
gap> AutomorphismsOfTable( m );
Group([ (2,3) ])
gap> Display( m );
2.A6.2_1mod2
A = E(3)-E(3)^2
= Sqrt(-3) = i3
B = -E(12)^7+E(12)^11
= Sqrt(3) = r3
]]></Example>
<P/>
When considering several characteristics in parallel, one argues as follows.
The possible class fusions from a Brauer table <M>mtbl</M> to its
ordinary table <M>tbl</M> are given by the orbit of a fixed class fusion
under the action of the table automorphisms of <M>tbl</M>.
If there are several orbits under the action of the automorphisms
of <M>mtbl</M> then we choose one orbit.
Due to this choice, only those table automorphisms of <M>tbl</M> are
admissible for other characteristics that stabilize the chosen orbit.
For the second characteristic, we take again the set of all class fusions
from the Brauer table to <M>tbl</M>, and split it into orbits under the
table automorphisms of the Brauer table.
Now there are two possibilities.
Either the action of the admissible subgroup of automorphisms of <M>tbl</M>
joins these orbits into one orbit or not.
In the former case, we choose again one of the orbits,
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