71.2 Access to Library Character Tables

If the only argument is a string tblname and if this is an admissible name (see below) of a library character table then CharacterTableFromLibrary returns this library table, otherwise fail.

If CharacterTableFromLibrary is called with more than one argument then the first must be a string series specifying a series of groups which is implemented via a generic character table, for example "Symmetric" for symmetric groups; the remaining arguments specialise then the desired member of the series (see Generic Character Tables for a list of available generic tables). If no generic table with name series is available or if the parameters are not admissible then CharacterTableFromLibrary returns fail.

A call of CharacterTableFromLibrary may cause to read some library files and to construct the table object from the data stored in these files, so fetching a library table may take more time than on expects.

CharacterTableFromLibrary is called by CharacterTable if the first argument is a string, so one may also call CharacterTable.

Admissible names for the ordinary character table t of the group G are

-

an ATLAS like name if t is an ATLAS table (see ATLAS Tables), for example "M22" for the table of the Mathieu group M₂₂, "L2(13).2" for L₂(13):2, and "12_1.U4(3).2_1" for 12₁\.U₄(3)\.2₁,

(The difference to the name printed in the ATLAS is that subscripts and superscripts are omitted except if they are used to qualify integer values, and double dots are replaced by a single dot.)

-

the names that were admissible for tables of G in sf CAS if the sf CAS table library contained a table of G, for example sl42 for the table of the alternating group A₈,

(But note that the GAP table may be different from that in sf CAS, see CAS Tables.)

-

some ``relative'' names, as follows.

If G is the n--th maximal subgroup (in decreasing group order) of a group whose library table s is available in GAP and stores the Maxes value (see Maxes), and if name is an admissible name for s then nameMn is admissible for t. For example, the name "J3M2" can be used to access the second maximal subgroup of the sporadic simple Janko group J₃ which has the admissible name J3.

If G is a nontrivial Sylow p normalizer in a sporadic simple group with admissible name name, --where nontrivial means that G is not isomorphic to a subgroup of p:(p-1)-- then nameNp is an admissible name of t. For example, the name "J4N11" can be used to access the table of the Sylow 11 normalizer in the sporadic simple Janko group J₄.

In a few cases, the table of the Sylow p subgroup of G is accessible via the name nameSylp where name is an admissible name of the table of G. For example, "A11Syl2" is an admissible name for the table of the Sylow 2 subgroup of the alternating group A₁₁.

In a few cases, the table of an element centralizer in G is accessible via the name nameCcl where name is an admissible name of the table of G. For example, "M11C2" is an admissible name for the table of an involution centralizer in the Mathieu group M₁₁.

The recommended way to access Brauer tables from the library is via the mod operator from the ordinary table and the desired characteristic (see BrauerTable, Operators for Character Tables), so it is not necessary to define admissible names of Brauer tables.

Generic character tables are accessible only by the name given by their Identifier value (see Generic Character Tables).

Case is not significant for character table names. For example, both "suzm3" and "SuzM3" are admissible names for the third maximal subgroup of the sporadic simple Suzuki group.

gap> s5:= CharacterTable( "A5.2" );
CharacterTable( "A5.2" )
gap> sym5:= CharacterTable( "Symmetric", 5 );
CharacterTable( "Sym(5)" )
gap> TransformingPermutationsCharacterTables( s5, sym5 );
rec( columns := (2,3,4,7,5), rows := (1,7,3,4,6,5,2), group := Group(()) )

gap> CharacterTable( "J5" );
fail
gap> CharacterTable( "A5" ) mod 2;
BrauerTable( "A5", 2 )

is a list of identifiers of the tables of all maximal subgroups of tbl. This is meaningful usually only for library tables, and there is no default method to compute the value.

If the Maxes value of tbl is stored then it lists exactly one representative for each conjugacy class of maximal subgroups of the group of tbl; the tables of these maximal subgroups are then available in the GAP table library, and the fusions to tbl are stored on these tables.

gap> tbl:= CharacterTable( "M11" );;
gap> HasMaxes( tbl );
true
gap> maxes:= Maxes( tbl );
[ "A6.2_3", "L2(11)", "3^2:Q8.2", "A5.2", "2.S4" ]
gap> CharacterTable( maxes[1] );
CharacterTable( "A6.2_3" )

Similar to group libraries (see Chapter Group Libraries), the GAP character table library can be used to search for ordinary character tables with prescribed properties.

A specific library table can be selected by an admissible name (see CharacterTableFromLibrary).

The selection function for character tables with certain abstract properties is AllCharacterTableNames. Contrary to the situation in the case of group libraries, the selection function returns a list not of library character tables but of their names; using CharacterTable one can then access the tables themselves.

AllCharacterTableNames takes an arbitrary even number of arguments. The argument at each odd position must be a function, and the argument at the subsequent even position must be a value that this function must return when called for the character table in question, in order to have the name of the table included in the selection, or a list of such values. For example,

gap> names:= AllCharacterTableNames();;

gap> simpnames:= AllCharacterTableNames( IsSimple, true );;
gap> AllCharacterTableNames( IsSimple, true, Size, [ 1 .. 100 ] );
[ "A5" ]

For the sake of efficiency, the arguments IsSimple and IsSporadicSimple followed by true are handled in a special way, GAP need not read all files of the table library in these cases in order to find the desired names.

If the function OfThose is an argument at an odd position then the following argument func must be a function that takes a character table and returns a name of a character table or a list of names; this is interpreted as replacement of the names computed up to this position by the union of names returned by func. For example, func may be Maxes (see Maxes) or NamesOfFusionSources (see NamesOfFusionSources).

gap> maxesnames:= AllCharacterTableNames( IsSporadicSimple, true,
>                                         HasMaxes, true,
>                                         OfThose, Maxes );;

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