67.3 Character Table Categories

Every ``character table like object'' in GAP lies in the category IsNearlyCharacterTable. There are four important subcategories, namely the ordinary tables in IsOrdinaryTable, the Brauer tables in IsBrauerTable, the union of these two in IsCharacterTable, and the incomplete ordinary tables in IsCharacterTableInProgress.

We want to distinguish ordinary and Brauer tables because a Brauer table may delegate tasks to the ordinary table of the same group, for example the computation of power maps. A Brauer table is constructed from an ordinary table and stores this table upon construction (see OrdinaryCharacterTable).

Furthermore, IsOrdinaryTable and IsBrauerTable denote character tables that provide enough information to compute all power maps and irreducible characters (and in the case of Brauer tables to get the ordinary table), for example because the underlying group (see UnderlyingGroup) is known or because the table is a library table (see Chapter The Character Table Library). We want to distinguish these tables from partially known ordinary tables that cannot be asked for all power maps or all irreducible characters.

The character table objects in IsCharacterTable are always immutable (see Mutability and Copyability). This means mainly that the ordering of conjugacy classes used for the various attributes of the character table cannot be changed; see Sorted Character Tables for how to compute a character table with a different ordering of classes.

The GAP objects in IsCharacterTableInProgress represent incomplete ordinary character tables. This means that not all irreducible characters, not all power maps are known, and perhaps even the number of classes and the centralizer orders are known. Such tables occur when the character table of a group G is constructed using character tables of related groups and information about G but for example without explicitly computing the conjugacy classes of G. An object in IsCharacterTableInProgress is first of all mutable, so nothing is stored automatically on such a table, since otherwise one has no control of side-effects when a hypothesis is changed. Operations for such tables may return more general values than for other tables, for example class functions may contain unknowns (see Chapter Unknowns) or lists of possible values in certain positions, the same may happen also for power maps and class fusions (see Parametrized Maps). @Incomplete tables in this sense are currently not supported and will be described in a chapter of their own when they become available.@ Note that the term ``incomplete table'' shall express that GAP cannot compute certain values such as irreducible characters or power maps. A table with access to its group is therefore always complete, also if its irreducible characters are not yet stored.

gap> g:= SymmetricGroup( 4 );;
gap> tbl:= CharacterTable( g );  modtbl:= tbl mod 2;
CharacterTable( Sym( [ 1 .. 4 ] ) )
BrauerTable( Sym( [ 1 .. 4 ] ), 2 )
gap> IsCharacterTable( tbl );  IsCharacterTable( modtbl );
true
true
gap> IsBrauerTable( modtbl );  IsBrauerTable( tbl );
true
false
gap> IsOrdinaryTable( tbl );  IsOrdinaryTable( modtbl );
true
false
gap> IsCharacterTable( g );  IsCharacterTable( Irr( g ) );
false
false

is the info class (see Info Functions) for computations with character tables.

Every character table like object lies in this family (see Families).

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