MultiplicityFree Permutation Characters in GAP
THOMAS BREUER
Lehrstuhl D für Mathematik
RWTH, 52056 Aachen, Germany
October 6th, 2000
This note shows a few examples of GAP computations concerning
multiplicityfree permutation characters,
with an emphasis on the classification of the faithful multiplicityfree
permutation characters of the sporadic simple groups and their automorphism
groups given in [BL96].
For examples on GAP computations with permutation characters in general,
see the note [Bre].
For further questions about GAP, consult its
Reference Manual;
in particular, for the description of the commands for character tables,
see the chapter "Character Tables".
Section 1 of this note shows how to interpret the individual
data available in the database.
In Section 2, the main idea is to gather information from
the database as a whole, by filtering items with suitable properties.
Finally, Section 3 gives an impression how GAP
can be used to obtain results such as the classification of described
in [BL96].
Contents
1 The Database of MultiplicityFree Characters
1.1 The Faithful MultiplicityFree Permutation Characters of M_{11}
1.2 The Faithful MultiplicityFree Permutation Characters of M_{12}.2
2 Using the Database
3 Using the Functions to Compute MultiplicityFree Permutation Characters
3.1 Using Tables of Marks
3.2 Dealing with Possible Permutation Characters
1 The Database of MultiplicityFree Characters
The database lists, for each group G that is either a sporadic simple
group or an automorphism group of a sporadic simple group,
a description of all conjugacy classes of subgroups H of G such that
the action of G on the right cosets of H is a faithful and
multiplicityfree permutation representation of G,
plus the permutation character of this representation.
The format how this information is stored is explained below,
subtleties such as possibly equal characters for different classes of
subgroups are discussed in Section 2.
(A GAP database providing more information about most of these
representations is in preparation;
this will cover, i.a., the character tables of the endomorphism rings of
these representations and the permutation representations themselves.)
The data is stored in the file multfree.dat,
which is part of the Character Table Library [Bre22] of the GAP
system [GAP04] as well as the file you are currently reading.
We load this GAP package and the data file into GAP 4.
Afterwards the function MultFreePermChars is available.
gap> LoadPackage( "ctbllib", false );
true
gap> if not IsBound( MultFreePermChars ) then
> ReadPackage( "ctbllib", "tst/multfree.dat" );

Loading the Database of MultiplicityFree Permutation Characters
of the Sporadic Simple Groups and Their Automorphism Groups,
by T. Breuer and K. Lux;
call `MultFreePermChars( <name> )' for accessing the data
for the group whose character table has identifier <name>.

> fi;
1.1 The Faithful MultiplicityFree Permutation Characters of
M_{11}
We start with the inspection of the Mathieu group M_{11},
as an example of a simple group that is dealt with in the database.
gap> info:= MultFreePermChars( "M11" );
[ rec( ATLAS := "1a+10a", character := Character( CharacterTable( "M11" ),
[ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ), group := "$M_{11}$", rank := 2,
subgroup := "$A_6.2_3$" ),
rec( ATLAS := "1a+10a+11a", character := Character( CharacterTable( "M11" ),
[ 22, 6, 4, 2, 2, 0, 0, 0, 0, 0 ] ), group := "$M_{11}$", rank := 3,
subgroup := "$A_6 \\leq A_6.2_3$" ),
rec( ATLAS := "1a+11a", character := Character( CharacterTable( "M11" ),
[ 12, 4, 3, 0, 2, 1, 0, 0, 1, 1 ] ), group := "$M_{11}$", rank := 2,
subgroup := "$L_2(11)$" ),
rec( ATLAS := "1a+11a+16ab+45a+55a",
character := Character( CharacterTable( "M11" ),
[ 144, 0, 0, 0, 4, 0, 0, 0, 1, 1 ] ), group := "$M_{11}$", rank := 6,
subgroup := "$11:5 \\leq L_2(11)$" ),
rec( ATLAS := "1a+10a+44a", character := Character( CharacterTable( "M11" ),
[ 55, 7, 1, 3, 0, 1, 1, 1, 0, 0 ] ), group := "$M_{11}$", rank := 3,
subgroup := "$3^2:Q_8.2$" ),
rec( ATLAS := "1a+10a+44a+55a",
character := Character( CharacterTable( "M11" ),
[ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ), group := "$M_{11}$", rank := 4,
subgroup := "$3^2:8 \\leq 3^2:Q_8.2$" ),
rec( ATLAS := "1a+10a+11a+44a",
character := Character( CharacterTable( "M11" ),
[ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0 ] ), group := "$M_{11}$", rank := 4,
subgroup := "$A_5.2$" ) ]
gap> List( info, x > x.rank );
[ 2, 3, 2, 6, 3, 4, 4 ]
gap> chars:= List( info, x > x.character );;
gap> degrees:= List( chars, x > x[1] );
[ 11, 22, 12, 144, 55, 110, 66 ]
We see that M_{11} has seven multiplicityfree permutation characters,
of the ranks and degrees listed above.
(Note that for multiplicityfree permutation characters,
the rank is equal to the number of irreducible constituents.)
More precisely, there are exactly seven conjugacy classes of subgroups of
M_{11} such that the permutation action on the cosets of these subgroups
is faithful and multiplicityfree.
For displaying the characters compatibly with the character table of M_{11},
we can use the Display operation.
Note that the column and row ordering of character tables in GAP
is compatible with that of the tables in the ATLAS of Finite Groups
([CCN^{+}85]).
gap> tbl:= CharacterTable( "M11" );
CharacterTable( "M11" )
gap> Display( tbl, rec( chars:= chars ) );
M11
2 4 4 1 3 . 1 3 3 . .
3 2 1 2 . . 1 . . . .
5 1 . . . 1 . . . . .
11 1 . . . . . . . 1 1
1a 2a 3a 4a 5a 6a 8a 8b 11a 11b
2P 1a 1a 3a 2a 5a 3a 4a 4a 11b 11a
3P 1a 2a 1a 4a 5a 2a 8a 8b 11a 11b
5P 1a 2a 3a 4a 1a 6a 8b 8a 11a 11b
11P 1a 2a 3a 4a 5a 6a 8a 8b 1a 1a
Y.1 11 3 2 3 1 . 1 1 . .
Y.2 22 6 4 2 2 . . . . .
Y.3 12 4 3 . 2 1 . . 1 1
Y.4 144 . . . 4 . . . 1 1
Y.5 55 7 1 3 . 1 1 1 . .
Y.6 110 6 2 2 . . 2 2 . .
Y.7 66 10 3 2 1 1 . . . .
The subgroup component of each record in info describes
the isomorphism type of a subgroup U of M_{11} such that the value π
of the character component is induced from the trivial character of U;
in other words, U is a point stabilizer of the permutation representation
of M_{11} with character π.
(Contrary to this example, in general it may happen that different classes of
subgroups induce the same permutation character,
and that these subgroups may also be nonisomorphic;
see Section 2 for details.)
gap> subgroups:= List( info, x > x.subgroup );
[ "$A_6.2_3$", "$A_6 \\leq A_6.2_3$", "$L_2(11)$", "$11:5 \\leq L_2(11)$",
"$3^2:Q_8.2$", "$3^2:8 \\leq 3^2:Q_8.2$", "$A_5.2$" ]
Each entry is a L^{A}T_{E}X format string that is either a name of the
point stabilizer or has the form <U> \leq <M> where <M> is the name
of a maximal subgroup containing the point stabilizer <U> as a proper
subgroup; in the former case, the point stabilizer is itself maximal.
Note that a backslash occurring in a subgroup string is escaped by another
backslash;
but only a single backslash is printed when the string is printed via
the function Print.
gap> Print( subgroups[2], "\n" );
$A_6 \leq A_6.2_3$
Finally, the ATLAS component of each record in info describes the
character value in terms of its irreducible constituents,
as is computed by the function PermCharInfo.
Examples can be found in Section 3;
for details about the output format,
see the documentation for this function in the GAP Reference Manual.
1.2 The Faithful MultiplicityFree Permutation Characters of
M_{12}.2
The automorphism group of a sporadic simple group G is either equal to G
or an upward extension of G by an outer automorphism of order 2.
The nonsimple automorphism group M_{12}.2 of the Mathieu group M_{12}
serves as an example of the latter situation.
In addition to the aspects mentioned in Section 1.1,
here we meet the situation that a permutation character either is induced
from a permutation character of M_{12} or extends such a
(not necessarily multiplicityfree) permutation character.
The former case occurs exactly if the corresponding point stabilizer lies in
M_{12}.
gap> info:= MultFreePermChars( "M12.2" );;
gap> Length( info );
13
gap> info[1];
rec( ATLAS := "1a^{\\pm}+11ab",
character := Character( CharacterTable( "M12.2" ),
[ 24, 0, 8, 6, 0, 4, 4, 0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
group := "$M_{12}.2$", rank := 3, subgroup := "$M_{11}$" )
gap> info[2];
rec( ATLAS := "1a^++11ab+55a^++66a^+",
character := Character( CharacterTable( "M12.2" ),
[ 144, 0, 16, 9, 0, 0, 4, 0, 1, 0, 0, 1, 12, 4, 0, 0, 2, 2, 0, 1, 1 ] ),
group := "$M_{12}.2$", rank := 4, subgroup := "$L_2(11).2$" )
The first character in the list info is induced from the trivial character
of a subgroup of type M_{11} inside M_{12},
the second character is induced from the trivial character of a L_{2}(11).2
subgroup whose intersection with M_{12} is of type L_{2}(11).
We can distinguish the two kinds of permutation characters by explicitly
using the character tables;
for example, a permutation character is induced from a subgroup of a normal
subgroup if and only if it vanishes outside the classes forming this
subgroup.
gap> m12:= CharacterTable( "M12" );;
gap> m122:= UnderlyingCharacterTable( info[1].character );;
gap> fus:= GetFusionMap( m12, m122 );
[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 12, 12 ]
gap> outer:= Difference( [ 1 .. NrConjugacyClasses( m122 ) ], fus );
[ 13, 14, 15, 16, 17, 18, 19, 20, 21 ]
gap> info[1].character{ outer };
[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> info[2].character{ outer };
[ 12, 4, 0, 0, 2, 2, 0, 1, 1 ]
A perhaps easier way is to look at the ATLAS components of the info
records.
Namely, the characters induced from subgroups of M_{12} have both
linear characters of M_{12}.2 as constituents,
which is expressed by the substring "1a^{\\pm}".
More generally, the ATLAS component lists the irreducible constituents
of the restriction to M_{12}, where the two extensions of a character
to M_{12}.2 are distinguished by a superscript +, −, or ±;
the latter means that both extensions occur.
The ATLAS components describing the constituents relative to a subgroup
of index 2 can be computed using the GAP function
PermCharInfoRelative, see Section 3.
It should be noted that the \leq substrings in the subgroup component
cannot be used to distinguish the two kinds of permutation characters,
since these substrings refer only to maximal subgroups different from
M_{12}.
Examples are the first entry in info (see above), the fourth entry
(containing a character that is induced from a subgroup of type A_{6}.2_{2}
which lies in a A_{6}.2^{2} subgroup that is maximal in M_{11}),
and the nineth entry (containing a character induced from a subgroup of
index 2 in a (2^{2} ×A_{5}).2 subgroup that is maximal in M_{12}.2.
gap> info[4];
rec( ATLAS := "1a^{\\pm}+11ab+54a^{\\pm}+66a^{\\pm}",
character := Character( CharacterTable( "M12.2" ),
[ 264, 24, 24, 12, 0, 4, 4, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
group := "$M_{12}.2$", rank := 7, subgroup := "$A_6.2_2 \\leq A_6.2^2$" )
gap> info[9];
rec(
ATLAS := "1a^++16ab+45a^++54a^{\\pm}+55a^+66a^{\\pm}+99a^+144a^++176a^",
character := Character( CharacterTable( "M12.2" ),
[ 792, 32, 24, 0, 6, 0, 2, 2, 0, 0, 2, 0, 0, 0, 8, 0, 0, 0, 2, 0, 0 ] ),
group := "$M_{12}.2$", rank := 11,
subgroup := "$(2 \\times A_5).2 \\leq (2^2 \\times A_5).2$" )
2 Using the Database
In this section, we study the complete list of multiplicityfree
permutation characters of the sporadic simple groups and their
automorphism groups as a whole.
gap> info:= MultFreePermChars( "all" );;
gap> Length( info );
267
gap> Length( Set( info ) );
262
gap> chars:= List( info, x > x.character );;
gap> Length( Set( chars ) );
261
We see that there are exactly 267 conjugacy classes of subgroups
such that the permutation representation on the cosets is multiplicityfree.
Only 262 of the info records are different,
and there is exactly one case where two different info records belong to
the same permutation character.
Let us look where these multiple entries arise.
gap> distrib:= List( info, x > Position( chars, x.character ) );;
gap> ambiguous:= Filtered( InverseMap( distrib ), IsList );
[ [ 12, 15 ], [ 40, 41 ], [ 83, 84 ], [ 88, 90 ], [ 132, 133 ], [ 202, 203 ] ]
gap> except:= Filtered( ambiguous, x > info[ x[1] ] <> info[ x[2] ] );
[ [ 83, 84 ] ]
gap> ambiguous:= Difference( ambiguous, except );;
gap> info{ except[1] };
[ rec( ATLAS := "1a+22a+230a",
character := Character( CharacterTable( "M23" ),
[ 253, 29, 10, 5, 3, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0 ] ),
group := "$M_{23}$", rank := 3, subgroup := "$L_3(4).2_2$" ),
rec( ATLAS := "1a+22a+230a",
character := Character( CharacterTable( "M23" ),
[ 253, 29, 10, 5, 3, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0 ] ),
group := "$M_{23}$", rank := 3, subgroup := "$2^4:A_7$" ) ]
So the Mathieu group M_{23} contains two classes of maximal subgroups,
of the structures L_{3}(4).2_{2} and 2^{4}:A_{7}, respectively,
such that the characters of the permutation representations on the
cosets of these subgroups are equal.
Furthermore, it is a consequence of the classification in [BL96]
that in all cases except this one,
the isomorphism types of the point stabilizers are uniquely determined
by the permutation characters.
gap> ambiginfo:= info{ List( ambiguous, x > x[1] ) };;
gap> for pair in ambiginfo do
> Print( pair.group, ", ", pair.subgroup, ", ", pair.ATLAS, "\n" );
> od;
$M_{12}$, $A_6.2_1 \leq A_6.2^2$, 1a+11ab+54a+55a
$M_{22}$, $A_7$, 1a+21a+154a
$HS$, $U_3(5).2$, 1a+175a
$McL$, $M_{22}$, 1a+22a+252a+1750a
$Fi_{22}$, $O_7(3)$, 1a+429a+13650a
In the other five cases of ambiguities, the whole info records are
equal, and from the above list we conclude that for each pair,
the point stabilizers are isomorphic.
In fact the subgroups are conjugate in the outer automorphism groups
of the simple groups involved.
Next let us look at the distribution of ranks.
gap> Collected( List( info, x > x.rank ) );
[ [ 2, 11 ], [ 3, 31 ], [ 4, 25 ], [ 5, 43 ], [ 6, 24 ], [ 7, 21 ],
[ 8, 26 ], [ 9, 16 ], [ 10, 17 ], [ 11, 9 ], [ 12, 9 ], [ 13, 8 ],
[ 14, 4 ], [ 15, 3 ], [ 16, 3 ], [ 17, 5 ], [ 18, 5 ], [ 19, 2 ],
[ 20, 2 ], [ 23, 1 ], [ 26, 1 ], [ 34, 1 ] ]
gap> max:= Filtered( info, x > x.rank = 34 );;
gap> max[1].group; max[1].subgroup; max[1].character[1];
"$F_{3+}.2$"
"$O_{10}^(2) \\leq O_{10}^(2).2$"
100354720284
The maximal rank, 34, is attained for a degree 100 354 720 284
character of F_{3+}.2 = Fi_{24}.
For the nonsimple automorphism groups of sporadic simple groups,
the simple group G involved is of index 2,
and each permutation characters either is induced from a character of G
or extends a permutation character of G.
gap> nonsimple:= Filtered( info,
> x > not IsSimple( UnderlyingCharacterTable( x.character ) ) );;
gap> Length( nonsimple );
120
gap> ind:= Filtered( nonsimple, x > ScalarProduct( x.character,
> Irr( UnderlyingCharacterTable( x.character ) )[2] ) = 1 );;
gap> Length( ind );
48
There are exactly 120 multiplicityfree permutation characters of
nonsimple automorphism groups of sporadic simple groups,
and 48 of them are induced from characters of the simple groups.
(Note that the second irreducible character of the GAP character tables
in question is the unique nontrivial linear character.)
gap> ind[1];
rec( ATLAS := "1a^{\\pm}+11ab",
character := Character( CharacterTable( "M12.2" ),
[ 24, 0, 8, 6, 0, 4, 4, 0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
group := "$M_{12}.2$", rank := 3, subgroup := "$M_{11}$" )
gap> ForAll( ind, x > x.ATLAS{ [ 1 .. 8 ] } = "1a^{\\pm}" );
true
Another possibility to select the induced characters is to check whether
the initial part of the ATLAS component is the string "1a^{\\pm}".
3 Using the Functions to Compute MultiplicityFree Permutation
Characters
The functions MultFreeFromTomAndTable and MultFree will be used later on.
(The functions can also be found in the file multfree.g,
which can be downloaded from the same webpage where also this file can
be found.)
For a character table tbl for which the table of marks is available in
the GAP library,
the function MultFreeFromTomAndTable returns the list of all
multiplicityfree permutation characters of tbl.
gap> BindGlobal( "MultFreeFromTomAndTable", function( tbl )
> local tom, # the table of marks
> fus, # fusion map from `t' to `tom'
> perms; # perm. characters of `t'
>
> if HasFusionToTom( tbl ) or HasUnderlyingGroup( tbl ) then
> tom:= TableOfMarks( tbl );
> else
> Error( "no table of marks for character table <tbl> available" );
> fi;
> fus:= FusionCharTableTom( tbl, tom );
> if fus = fail then
> Error( "no unique fusion from <tbl> to the table of marks" );
> fi;
> perms:= PermCharsTom( tbl, tom );
> return Filtered( perms,
> x > ForAll( Irr( tbl ),
> y > ScalarProduct( tbl, x, y ) <= 1 ) );
> end );
TestPerm calls the GAP library functions TestPerm1, TestPerm2,
and TestPerm3; the return value is true if the argument pi is
a possible permutation character of the character table tbl,
and false otherwise.
gap> BindGlobal( "TestPerm", function( tbl, pi )
> return TestPerm1( tbl, pi ) = 0
> and TestPerm2( tbl, pi ) = 0
> and not IsEmpty( TestPerm3( tbl, [ pi ] ) );
> end );
Let H be a character table, S be a list of characters of H,
psi a character of H, scprS a matrix, the ith entry being the
coefficients of the decomposition of the induced character of S[i]
to a supergroup G, say, of H, scprpsi the decomposition of psi
induced to G, and k a positive integer.
CharactersInducingWithBoundedMultiplicity returns the list
C( S, psi, k );
this is the list of all those characters psi + ϑ of
multiplicity at most k such that all constituents of ϑ are
contained in S.
gap> DeclareGlobalFunction( "CharactersInducingWithBoundedMultiplicity" );
gap> InstallGlobalFunction( CharactersInducingWithBoundedMultiplicity,
> function( H, S, psi, scprS, scprpsi, k )
> local result, # the list $S( .. )$
> chi, # $\chi$
> scprchi, # decomposition of $\chi^G$
> i, # loop from `1' to `k'
> allowed, # indices of possible constituents
> Sprime, # $S^{\prime}_i$
> scprSprime; # decomposition of characters in $S^{\prime}_i$,
> # induced to $G$
>
> if IsEmpty( S ) then
>
> # Test whether `psi' is a possible permutation character.
> if TestPerm( H, psi ) then
> result:= [ psi ];
> else
> result:= [];
> fi;
>
> else
>
> # Fix a character $\chi$.
> chi := S[1];
> scprchi := scprS[1];
>
> # Form the union.
> result:= CharactersInducingWithBoundedMultiplicity( H,
> S{ [ 2 .. Length( S ) ] }, psi,
> scprS{ [ 2 .. Length( S ) ] }, scprpsi, k );
> for i in [ 1 .. k ] do
> allowed := Filtered( [ 2 .. Length( S ) ],
> j > Maximum( i * scprchi + scprS[j] ) <= k );
> Sprime := S{ allowed };
> scprSprime := scprS{ allowed };
>
> Append( result, CharactersInducingWithBoundedMultiplicity( H,
> Sprime, psi + i * chi,
> scprSprime, scprpsi + i * scprchi, k ) );
> od;
>
> fi;
>
> return result;
> end );
Let G and H be character tables of groups G and H, respectively,
such that H is a subgroup of G and the class fusion from H to G
is stored on H.
MultAtMost returns the list of all characters ϕ^{G} of G
of multiplicity at most k such that ϕ is a possible permutation
character of H.
gap> BindGlobal( "MultAtMost", function( G, H, k )
> local triv, # $1_H$
> permch, # $(1_H)^G$
> scpr1H, # decomposition of $(1_H)^G$
> rat, # rational irreducible characters of $H$
> ind, # induced rational irreducible characters
> mat, # decomposition of `ind'
> allowed, # indices of possible constituents
> S0, # $S_0$
> scprS0, # decomposition of characters in $S_0$,
> # induced to $G$, with $Irr(G)$
> cand; # list of multiplicityfree candidates, result
>
> # Compute $(1_H)^G$ and its decomposition into irreducibles of $G$.
> triv := TrivialCharacter( H );
> permch := Induced( H, G, [ triv ] );
> scpr1H := MatScalarProducts( G, Irr( G ), permch )[1];
>
> # If $(1_H)^G$ has multiplicity larger than `k' then we are done.
> if Maximum( scpr1H ) > k then
> return [];
> fi;
>
> # Compute the set $S_0$ of all possible nontrivial
> # rational constituents of a candidate of multiplicity at most `k',
> # that is, all those rational irreducible characters of
> # $H$ that induce to $G$ with multiplicity at most `k'.
> rat:= RationalizedMat( Irr( H ) );
> ind:= Induced( H, G, rat );
> mat:= MatScalarProducts( G, Irr( G ), ind );
> allowed:= Filtered( [ 1.. Length( mat ) ],
> x > Maximum( mat[x] + scpr1H ) <= k );
> S0 := rat{ allowed };
> scprS0 := mat{ allowed };
>
> # Compute $C( S_0, 1_H, k )$.
> cand:= CharactersInducingWithBoundedMultiplicity( H,
> S0, triv, scprS0, scpr1H, k );
>
> # Induce the candidates to $G$, and return the sorted list.
> cand:= Induced( H, G, cand );
> Sort( cand );
> return cand;
> end );
MultFree returns MultAtMost( G, H, 1 ).
gap> BindGlobal( "MultFree", function( G, H )
> return MultAtMost( G, H, 1 );
> end );
Let tbl be a character table with known Maxes value,
and k a positive integer.
The function PossiblePermutationCharactersWithBoundedMultiplicity
returns a record with the following components.
identifier
the Identifier value of tbl,
maxnames
the list of names of the maximal subgroups of tbl,
permcand
at the ith position the list of those possible permutation
characters of tbl whose multiplicity is at most k
and which are induced from the ith maximal subgroup of tbl,
and
k
the given bound k for the multiplicity.
gap> BindGlobal( "PossiblePermutationCharactersWithBoundedMultiplicity",
> function( tbl, k )
> local permcand, # list of all mult. free perm. character candidates
> maxname, # loop over tables of maximal subgroups
> max; # one table of a maximal subgroup
>
> if not HasMaxes( tbl ) then
> return fail;
> fi;
>
> permcand:= [];
>
> # Loop over the tables of maximal subgroups.
> for maxname in Maxes( tbl ) do
>
> max:= CharacterTable( maxname );
> if max = fail or GetFusionMap( max, tbl ) = fail then
>
> Print( "#E no fusion `", maxname, "' > `", Identifier( tbl ),
> "' stored\n" );
> Add( permcand, Unknown() );
>
> else
>
> # Compute the possible perm. characters inducing through `max'.
> Add( permcand, MultAtMost( tbl, max, k ) );
>
> fi;
> od;
>
> # Return the result record.
> return rec( identifier := Identifier( tbl ),
> maxnames := Maxes( tbl ),
> permcand := permcand,
> k := k );
> end );
3.1 Using Tables of Marks
As a small example for the computation of multiplicityfree permutation
characters from the table of marks of a group, we consider the alternating
group A_{5}.
Its character table as well as its table of marks are accessible from the
respective GAP library, via the identifier A5.
gap> tbl:= CharacterTable( "A5" );;
gap> chars:= MultFreeFromTomAndTable( tbl );
[ Character( CharacterTable( "A5" ), [ 12, 0, 0, 2, 2 ] ),
Character( CharacterTable( "A5" ), [ 10, 2, 1, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 6, 2, 0, 1, 1 ] ),
Character( CharacterTable( "A5" ), [ 5, 1, 2, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ) ]
As the GAP databases do not provide information about the isomorphism
types of arbitrary subgroups, there is no way to compute automatically the
subgroup strings as contained in the database of multiplicityfree
permutation characters (cf. Section 1).
Of course it is easy to see that the above characters of A_{5} are induced
from the trivial characters of the cyclic group of order 5,
the dihedral groups of orders 6 and 10, the alternating group A_{4},
and the group A_{5} itself, respectively.
The ATLAS information used in the database records can be computed
using the GAP function PermCharInfo.
gap> PermCharInfo( tbl, chars ).ATLAS;
[ "1a+3ab+5a", "1a+4a+5a", "1a+5a", "1a+4a", "1a" ]
As an example for a nonsimple group, we repeat the computation of
all multiplicityfree permutation characters of M_{12}.2,
using the GAP table of marks.
gap> tbl:= CharacterTable( "M12.2" );;
gap> chars:= MultFreeFromTomAndTable( tbl );;
gap> lib:= MultFreePermChars( "M12.2" );;
gap> Length( lib ); Length( chars );
13
15
gap> Difference( chars, List( lib, x > x.character ) );
[ Character( CharacterTable( "M12.2" ),
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
Character( CharacterTable( "M12.2" ),
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
This confirms the classification for M_{12}.2, since the additional
characters found from the table of marks are not faithful.
The corresponding ATLAS information is computed using the GAP function
PermCharInfoRelative, since the constituents shall be listed relative to
the simple group M_{12}.
gap> tblsimple:= CharacterTable( "M12" );;
gap> PermCharInfoRelative( tblsimple, tbl, chars ).ATLAS;
[ "1a^++16ab+45a^+54a^{\\pm}+55a^{\\pm}bc+66a^++99a^{\\pm}+144a^++176a^+",
"1a^++11ab+45a^+54a^{\\pm}+55a^++66a^{\\pm}+99a^+120a^{\\pm}+144a^{\\pm}",
"1a^{\\pm}+11ab+45a^{\\pm}+54a^{\\pm}+55a^{\\pm}bc+99a^{\\pm}+120a^{\\pm}",
"1a^++16ab+45a^++54a^{\\pm}+55a^+66a^{\\pm}+99a^+144a^++176a^",
"1a^++16ab+45a^+54a^{\\pm}+66a^++99a^+144a^+",
"1a^++11ab+54a^{\\pm}+55a^++66a^++99a^+144a^+",
"1a^{\\pm}+11ab+54a^{\\pm}+55a^{\\pm}+99a^{\\pm}",
"1a^++16ab+45a^++54a^{\\pm}+66a^++144a^+",
"1a^{\\pm}+11ab+54a^{\\pm}+66a^{\\pm}", "1a^++16ab+45a^++66a^+",
"1a^++11ab+55a^++66a^+", "1a^{\\pm}+11ab+54a^{\\pm}", "1a^{\\pm}+11ab",
"1a^{\\pm}", "1a^+" ]
For more information about tables of marks, see [Pfe97].
3.2 Dealing with Possible Permutation Characters
In this section, we deal with possible permutation characters,
that is, characters that have certain properties of permutation
characters but for which no subgroups need to exist from whose trivial
characters they are induced.
For more information about such characters, see the section
"Possible Permutation Characters" in the GAP Reference Manual,
the paper [BP98], and the note [Bre].
We can compute possible permutation characters from the character table
of the group in question, the table of marks need not be available.
The problem is of course that for classifying the permutation characters,
we have to decide which of the candidates are in fact permutation
characters.
Here we show only two small examples that could also be handled via
tables of marks.
(The GAP code shown uses only standard functions lists, such as
List, Filtered, and ForAll, and functions for character tables,
such as Irr and ScalarProduct;
if you are not familiar with these functions, consult the corresponding
sections in the GAP Reference Manual.)
The first example is the Mathieu group M_{11} that has been inspected
already in Section 1.1.
This group is small enough for the computation of all possible permutation
characters, and then filtering out the multiplicityfree ones.
gap> tbl:= CharacterTable( "M11" );;
gap> perms:= PermChars( tbl );;
gap> multfree:= Filtered( perms,
> x > ForAll( Irr( tbl ), chi > ScalarProduct( chi, x ) <= 1 ) );
[ Character( CharacterTable( "M11" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
Character( CharacterTable( "M11" ), [ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 12, 4, 3, 0, 2, 1, 0, 0, 1, 1 ] ),
Character( CharacterTable( "M11" ), [ 22, 6, 4, 2, 2, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 55, 7, 1, 3, 0, 1, 1, 1, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 144, 0, 0, 0, 4, 0, 0, 0, 1, 1 ] ) ]
gap> Length( multfree );
8
Comparing this list with the seven faithful multiplicityfree permutation
characters of M_{11} shown in Section 1.1,
we see that all candidates are in fact permutation characters.
Without this information, we have to show, for each candidate,
the existence of a subgroup that serves as the point stabilizer.
Additionally, if we are interested in the subgroup information contained in
the database (cf. the subgroup components of the info records in
Section 1), we want to relate the point stabilizers to the
maximal subgroups of M_{11}.
In the case of the sporadic simple groups and their automorphism groups,
we can use the fact that for many of these groups,
the character tables of all maximal subgroups and the class fusions of these
tables are known.
Since each multiplicityfree permutation character of a group is either
trivial or induced from a multiplicityfree permutation character of a
maximal subgroup, we can thus reduce our problem to the computation of
multiplicityfree possible permutation characters of all maximal subgroups.
(That this really is a reduction can be read in [BL96].)
This approach is implemented in the function MultFree.
gap> tbl:= CharacterTable( "M11" );
CharacterTable( "M11" )
gap> maxes:= Maxes( tbl );
[ "A6.2_3", "L2(11)", "3^2:Q8.2", "A5.2", "2.S4" ]
gap> name:= maxes[1];;
gap> MultFree( tbl, CharacterTable( name ) );
[ Character( CharacterTable( "M11" ), [ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 22, 6, 4, 2, 2, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ) ]
The function MultFree computes all multiplicityfree characters of
the given character table that are induced from possible permutation
characters of the given character table of a subgroup.
(Note that these characters need not necessarily be faithful.)
If we loop over all classes of maximal subgroups then we get all
candidates for M_{11},
together with the information in which maximal subgroup the hypothetical
point stabilizer lies.
gap> cand:= [];;
gap> for name in maxes do
> max:= CharacterTable( name );
> Append( cand, List( MultFree( tbl, max ),
> chi > [ name, Size( tbl ) / Size( max ), chi ] ) );
> od;
gap> cand;
[ [ "A6.2_3", 11, Character( CharacterTable( "M11" ),
[ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ) ],
[ "A6.2_3", 11, Character( CharacterTable( "M11" ),
[ 22, 6, 4, 2, 2, 0, 0, 0, 0, 0 ] ) ],
[ "A6.2_3", 11, Character( CharacterTable( "M11" ),
[ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ) ],
[ "L2(11)", 12, Character( CharacterTable( "M11" ),
[ 12, 4, 3, 0, 2, 1, 0, 0, 1, 1 ] ) ],
[ "L2(11)", 12, Character( CharacterTable( "M11" ),
[ 144, 0, 0, 0, 4, 0, 0, 0, 1, 1 ] ) ],
[ "3^2:Q8.2", 55, Character( CharacterTable( "M11" ),
[ 55, 7, 1, 3, 0, 1, 1, 1, 0, 0 ] ) ],
[ "3^2:Q8.2", 55, Character( CharacterTable( "M11" ),
[ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ) ],
[ "A5.2", 66, Character( CharacterTable( "M11" ),
[ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0 ] ) ] ]
gap> Length( cand ); Length( Set( cand, x > x[3] ) );
8
7
We immediately see that the candidates of degrees 11, 12, 55, and 66
are permutation characters,
since they are obtained by inducing the trivial characters of the
maximal subgroups.
The permutation characters of degrees 22 and 144 can be established
in two steps.
First we note that the group A_{6}.2_{3} contains the subgroup A_{6} of index
2,
and the group L_{2}(11) contains a class of subgroups of index 12,
of isomorphism type 11:5.
Second the possible permutation characters of degrees 2 and 12
of these maximal subgroups of M_{11} are uniquely determined,
and inducing these characters to M_{11} yields in fact multiplicityfree
characters.
gap> max1:= CharacterTable( maxes[1] );;
gap> perms1:= PermChars( max1, [ 2 ] );
[ Character( CharacterTable( "A6.2_3" ), [ 2, 2, 2, 2, 2, 0, 0, 0 ] ) ]
gap> perms1[1]^tbl = cand[2][3];
true
gap> max2:= CharacterTable( maxes[2] );;
gap> perms2:= PermChars( max2, [ 12 ] );
[ Character( CharacterTable( "L2(11)" ), [ 12, 0, 0, 2, 2, 0, 1, 1 ] ) ]
gap> perms2[1]^tbl = cand[5][3];
true
The last candidate to deal with is the degree 110 character,
which might be induced from a subgroup of A_{6}.2_{3} or 3^{2}:Q_{8}.2
or both.
Let us first look at the possible permutation characters of degree 10
of A_{6}.2_{3}.
gap> PermChars( max1, [ 10 ] );
[ Character( CharacterTable( "A6.2_3" ), [ 10, 2, 1, 2, 0, 0, 2, 2 ] ),
Character( CharacterTable( "A6.2_3" ), [ 10, 2, 1, 2, 0, 2, 0, 0 ] ) ]
gap> OrdersClassRepresentatives( max1 );
[ 1, 2, 3, 4, 5, 4, 8, 8 ]
There are two possibilities, and only the first induces the candidate of
degree 110.
The latter follows from the fact that the nonzero character value of the
candidate on classes of element order 8 means that the
hypothetical point stabilizer contains elements of order 8,
cf. the Display call in Section 1.1.
The group A_{6}.2_{3} has a unique class of subgroups of index 10,
which are the Sylow 3 normalizers, of type 3^{2}:Q_{8}.
Since Q_{8} has no elements of order 8,
the first candidate is not a permutation character.
The remaining subgroup from which the degree 110 character can be induced
is 3^{2}:Q_{8}.2;
this group has three index 2 subgroups, and the candidate is in fact
induced from the trivial character of one of these subgroups.
gap> max3:= CharacterTable( maxes[3] );;
gap> classes:= SizesConjugacyClasses( max3 );;
gap> Filtered( ClassPositionsOfNormalSubgroups( max3 ),
> x > Sum( classes{ x } ) = Size( max3 ) / 2 );
[ [ 1, 2, 4, 5, 6 ], [ 1, 2, 3, 4, 5, 7 ], [ 1, 2, 4, 5, 8, 9 ] ]
gap> perms3:= PermChars( max3, [ 2 ] );
[ Character( CharacterTable( "3^2:Q8.2" ), [ 2, 2, 0, 2, 2, 0, 0, 2, 2 ] ),
Character( CharacterTable( "3^2:Q8.2" ), [ 2, 2, 0, 2, 2, 2, 0, 0, 0 ] ),
Character( CharacterTable( "3^2:Q8.2" ), [ 2, 2, 2, 2, 2, 0, 2, 0, 0 ] ) ]
gap> induced:= List( perms3, x > x^tbl );
[ Character( CharacterTable( "M11" ), [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 110, 6, 2, 6, 0, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 110, 14, 2, 2, 0, 2, 0, 0, 0, 0 ] ) ]
gap> Position( induced, cand[3][3] );
1
Putting these considerations together, we thus get a confirmation of the
classification for M_{11}.
As a second example, we look at the group M_{12}.2.
The database contains 13 characters,
and the approach using MultFree yields 17 different characters.
We are interested in disproving the candidates that are not
permutation characters.
gap> info:= MultFreePermChars( "M12.2" );;
gap> perms:= Set( List( info, x > x.character ) );;
gap> Length( info ); Length( perms );
13
13
gap> tbl:= CharacterTable( "M12.2" );;
gap> maxes:= Maxes( tbl );
[ "M12", "L2(11).2", "M12.2M3", "(2^2xA5):2", "2^3.(S4x2)", "4^2:D12.2",
"3^(1+2):D8", "S4xS3", "A5.2" ]
gap> cand:= [];;
gap> for name in maxes do
> max:= CharacterTable( name );
> Append( cand, List( MultFree( tbl, max ),
> chi > [ name, Size( tbl ) / Size( max ), chi ] ) );
> od;
gap> Length( cand ); Length( Set( List( cand, x > x[3] ) ) );
25
17
gap> toexclude:= Set( Filtered( cand, x > not x[3] in perms ) );
[ [ "M12", 2, Character( CharacterTable( "M12.2" ),
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ],
[ "M12", 2, Character( CharacterTable( "M12.2" ),
[ 440, 0, 24, 8, 8, 8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
, [ "M12", 2, Character( CharacterTable( "M12.2" ),
[ 1320, 0, 8, 6, 0, 8, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
, [ "M12", 2, Character( CharacterTable( "M12.2" ),
[ 1320, 0, 24, 6, 0, 4, 0, 0, 6, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] )
] ]
Clearly the degree 2 character is a permutation character,
but as it is not faithful, it is not contained in the database.
The other three characters are all induced from candidates of the maximal
subgroup M_{12},
and we may use the same approach for M_{12} in order to find out whether
they can be permutation characters.
gap> m12:= CharacterTable( "M12" );;
gap> subcand:= [];;
gap> submaxes:= Maxes( m12 );
[ "M11", "M12M2", "A6.2^2", "M12M4", "L2(11)", "3^2.2.S4", "M12M7", "2xS5",
"M8.S4", "4^2:D12", "A4xS3" ]
gap> for name in submaxes do
> max:= CharacterTable( name );
> Append( subcand, MultFree( m12, max ) );
> od;
gap> induced:= List( subcand, x > x^tbl );;
gap> Intersection( induced, List( toexclude, x > x[3] ) );
[ ]
Thus none of the candidates in the list toexclude is a permutation
character.
References
 [BL96]

T. Breuer and K. Lux, The multiplicityfree permutation characters of the
sporadic simple groups and their automorphism groups, Comm. Algebra
24 (1996), no. 7, 22932316. MR 1390375 (97c:20020)
 [BP98]

T. Breuer and G. Pfeiffer, Finding possible permutation characters, J.
Symbolic Comput. 26 (1998), no. 3, 343354. MR 1633876
(99e:20005)
 [Bre]

T. Breuer, Permutation Characters in GAP, https://www.math.rwthaachen.de/
~Thomas.Breuer/
ctbllib/doc2/
manual.pdf.
 [Bre22]

, The GAP Character Table Library, Version
1.3.3, https://www.math.rwthaachen.de/
~Thomas.Breuer/
ctbllib, Jan 2022, GAP package.
 [CCN^{+}85]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,
Atlas of finite groups, Oxford University Press, Eynsham, 1985,
Maximal subgroups and ordinary characters for simple groups, With
computational assistance from J. G. Thackray. MR 827219 (88g:20025)
 [GAP04]

The GAP Group, GAPGroups, Algorithms, and
Programming, Version 4.4, 2004, http://www.gapsystem.org.
 [Pfe97]

G. Pfeiffer, The subgroups of M_{24}, or how to compute the table of
marks of a finite group, Experiment. Math. 6 (1997), no. 3,
247270. MR 1481593 (98h:20032)
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