Besides the characters, powermaps are an important part of a character table. Often their computation is not easy, and in general they cannot be obtained from the matrix of irreducible characters, so it is useful to store them on the table.
If not only a single table is considered but different tables of groups and subgroups are used, also subgroup fusion maps must be known to get informations about the embedding or simply to induce or restrict characters.
These are examples of class functions which are called maps for short;
in GAP, maps are lists: Characters are maps, the lists of element orders,
centralizer orders, classlengths are maps, and for a permutation perm of
ListPerm( perm ) is a map.
When maps are constructed, in most cases one only knows that the image of any class is contained in a set of possible images, e.g. that the image of a class under a subgroup fusion is in the set of all classes with the same element order. Using further informations, like centralizer orders, powermaps and the restriction of characters, the sets of possible images can be diminished. In many cases, at the end the images are uniquely determined.
For this, many functions do not only work with maps but with
parametrized maps (or short paramaps): These are lists whose entries are
either the images themselves (i.e. integers for fusion maps, powermaps,
element orders etc. and cyclotomics for characters) or lists of possible
images. Thus maps are special paramaps. A paramap paramap can be
identified with the set of all maps map with
map[i] = paramap[i]
map[i] contained in
paramap[i]; we say that map is contained
in paramap then.
The composition of two paramaps is defined as the paramap that contains all compositions of elements of the paramaps. For example, the indirection of a character by a parametrized subgroup fusion map is the parametrized character that contains all possible restrictions of that character.
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