Generic character tables provide a means for writing down the character tables of all groups in a (usually infinite) series of similar groups, e.g. the cyclic groups, the symmetric groups or the general linear groups {rm GL}(2,q).
Let {G_q|q in I}, where I is an index set, be such a series. The table of a member G_q could be computed using a program for this series which takes q as parameter, and constructs the table. It is, however, desirable to compute not only the whole table but to get a single character or just one character value without computation the table. E.g. both conjugacy classes and irreducible characters of the symmetric group S_n are in bijection with the partitions of n. Thus for given n, it makes sense to ask for the character corresponding to a particular partition, and its value at a partition:
gap> t:= CharTable( "Symmetric" );;
gap> t.irreducibles[1][1]( 5, [ 3, 2 ], [ 2, 2, 1 ] );
1 # a character value of $S_5$
gap> t.orders[1]( 5, [ 2, 1, 1, 1 ] );
2 # a representative order in $S_5$
Generic table in GAP means that such local evaluation is possible, so GAP can also deal with tables that are too big to be computed as a whole. In some cases there are methods to compute the complete table of small members G_q faster than local evaluation. If such an algorithm is part of the generic table, it will be used when the generic table is used to compute the whole table (see CharTableSpecialized).
While the numbers of conjugacy classes for the series are usually not bounded, there is a fixed finite number of types (equivalence classes) of conjugacy classes; very often the equivalence relation is isomorphism of the centralizer of the representatives.
For each type t of classes and a fixed q in I, a parametrisation of the classes in t is a function that assigns to each conjugacy class of G_q in t a parameter by which it is uniquely determined. Thus the classes are indexed by pairs (t,p_t) for a type t and a parameter p_t for that type.
There has to be a fixed number of types of irreducibles characters of G_q, too. Like the classes, the characters of each type are parametrised.
In GAP, the parametrisations of classes and characters of the generic
table is given by the record fields classparam and charparam; they
are both lists of functions, each function representing the
parametrisation of a type. In the specialized table, the field
classparam contains the lists of class parameters, the character
parameters are stored in the field charparam of the irredinfo records
(see Character Table Records).
The centralizer orders, representative orders and all powermaps of the generic character table can be represented by functions in q, t and p_t; in GAP, however, they are represented by lists of functions in q and a class parameter where each function represents a type of classes. The value of a powermap at a particular class is a pair consisting of type and parameter that specifies the image class.
The values of the irreducible characters of G_q can be represented by functions in q, class type and parameter, character type and parameter; in GAP, they are represented by lists of lists of functions, each list of functions representing the characters of a type, the function (in q, character parameters and class parameters) representing the classes of a type in these characters.
Any generic table is a record like an ordinary character table (see Character Table Records). There are some fields which are used for generic tables only:
isGenericTable:true
specializedname:
domain:true if its argument is a valid q for G_q
in the series
wholetable:The table of G_q can be constructed by specializing q and evaluating the functions in the generic table (see CharTableSpecialized and the examples given in Examples of Generic Character Tables).
Contents of the Table Libraries and CharTable.
GAP 3.4.4