66.12 The Interface between Tables of Marks and Character Tables

Let tbl be the ordinary character table of the group G, and tom the table of marks of G. FusionCharTableTom determines the fusion of the classes of elements from tbl to the classes of cyclic subgroups on tom, that is, a list that contains at position i the position of the class of cyclic subgroups in tom that are generated by elements in the i-th conjugacy class of elements in tbl.

Three cases are handled differently.

1.

The fusion is explicitly stored on tbl. Then nothing has to be done. This happens only if both tbl and tom are tables from the GAP library (see The Character Table Library, The Library of Tables of Marks).

2.

The UnderlyingGroup values of tbl and tom are known and equal. Then the group is used to compute the fusion.

3.

There is neither fusion nor group information available. In this case only necessary conditions can be checked, and if they are not sufficient to detemine the fusion uniquely then fail is returned by FusionCharTableTom.

PossibleFusionsCharTableTom computes the list of possible fusions from tbl to tom, according to the criteria that have been checked. So if FusionCharTableTom returns a unique fusion then the list returned by PossibleFusionsCharTableTom for the same arguments contains exactly this fusion, and if FusionCharTableTom returns fail then the length of this list is different from 1. The optional argumanet quick, a boolean, indicates whether a unique fusion shall be returned as soon as it has been determined, without further checks. The default value of quick is false.

gap> a5c:= CharacterTable( "A5" );;
gap> fus:= FusionCharTableTom( a5c, a5 );
[ 1, 2, 3, 5, 5 ]

PermCharsTom returns the list of transitive permutation characters from the table of marks tom. In the first form, fus must be the fusion map from the ordinary character table of the group of tom to tom (see FusionCharTableTom). In the second form, tbl must be the character table of the group of which tom is the table of marks. If the fusion map is not uniquely determined (see FusionCharTableTom) then fail is returned.

If the fusion map fus is given as first argument then each transitive permutation character is represented by its values list. If the character table tbl is given then the permutation characters are class function objects (see Chapter Class Functions).

gap> PermCharsTom( a5c, a5 );
[ Character( CharacterTable( "A5" ), [ 60, 0, 0, 0, 0 ] ), 
  Character( CharacterTable( "A5" ), [ 30, 2, 0, 0, 0 ] ), 
  Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ), 
  Character( CharacterTable( "A5" ), [ 15, 3, 0, 0, 0 ] ), 
  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 ] ) ]
gap> PermCharsTom( fus, a5 )[1];
[ 60, 0, 0, 0, 0 ]

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