69.5 Subroutines for the Construction of Class Fusions

For two ordinary character tables subtbl and tbl, InitFusion returns a parametrized map that is a first approximation of the class fusion from subtbl to tbl, using condition 1. listed in the description of PossibleClassFusions (see PossibleClassFusions).

If there are classes for which no images are possible, according to this criterion, then fail is returned.

gap> subtbl:= CharacterTable( "2F4(2)" );;  tbl:= CharacterTable( "Ru" );;
gap> fus:= InitFusion( subtbl, tbl );
[ 1, 2, 2, 4, [ 5, 6 ], [ 5, 6, 7, 8 ], [ 5, 6, 7, 8 ], [ 9, 10 ], 11, 14, 
  14, [ 13, 14, 15 ], [ 16, 17 ], [ 18, 19 ], 20, [ 25, 26 ], [ 25, 26 ], 
  [ 5, 6 ], [ 5, 6 ], [ 5, 6 ], [ 5, 6, 7, 8 ], [ 13, 14, 15 ], 
  [ 13, 14, 15 ], [ 18, 19 ], [ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 
  [ 27, 28, 29 ], [ 27, 28, 29 ] ]

CheckPermChar replaces the entries of the parametrized map approxmap by improved values, according to condition 3. listed in the description of PossibleClassFusions (see PossibleClassFusions).

CheckPermChar returns true if no inconsistency occurred, and false otherwise.

gap> permchar:= Sum( Irr( tbl ){ [ 1, 5, 6 ] } );;
gap> CheckPermChar( subtbl, tbl, fus, permchar ); fus;
true
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20, 
  [ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ], [ 18, 19 ], 
  [ 25, 26 ], [ 25, 26 ], 27, 27 ]

ConsiderTableAutomorphisms replaces the entries of the parametrized map approxmap by improved values, according to condition 4. listed in the description of PossibleClassFusions (see PossibleClassFusions).

Afterwards exactly one representative of fusion maps (contained in approxmap) in each orbit under the action of the permutation group grp is contained in the modified parametrized map.

ConsiderTableAutomorphisms returns the list of positions where approxmap was changed.

gap> ConsiderTableAutomorphisms( fus, AutomorphismsOfTable( tbl ) );
[ 16 ]
gap> fus;
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20, 25, 
  [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ], [ 18, 19 ], [ 25, 26 ], 
  [ 25, 26 ], 27, 27 ]

Let subtbl and tbl be ordinary character tables, subchars and chars two lists of (values lists of) characters of subtbl and tbl, respectively, approxmap a parametrized map that is an approximation of the class fusion of subtbl in tbl, and parameters a record with components maxlen, minamb, maxamb (three integers), quick (a boolean), and contained (a function). Usual values of contained are ContainedCharacters or ContainedPossibleCharacters.

FusionsAllowedByResrictions replaces the entries of approxmap by improved values, according to condition 5. listed in the description of PossibleClassFusions (see PossibleClassFusions).

More precisely, the strategy used is as follows.

First, for each c Î chars , let restricted:= CompositionMaps( c, approxmap ).

-

If Indeterminateness( restricted ) = 1 and parameters.quick = false then the scalar products of restricted with subchars are checked; if not all scalar products are nonnegative integers then an empty list is returned, otherwise c is deleted from the list of characters to inspect.

-

Otherwise if Indeterminateness( minus ) is smaller than parameters.minamb then c is deleted from the list of characters.

-

If parameters.minamb £ Indeterminateness( restricted ) £ parameters.maxamb then construct poss:= parameters.contained( subtbl, subchars, restricted ) and Parametrized( poss ), and improve the approximation of the fusion map using UpdateMap.

If this yields no further immediate improvements then we branch. If there is a character from chars left with less or equal parameters.maxlen possible restrictions, compute the union of fusion maps allowed by these possibilities. Otherwise we choose a class C such that the possible restrictions of a character in chars differ at C, and compute recursively the union of all allowed fusion maps with image at C fixed in the set given by the current approximation of the fusion map.

gap> subtbl:= CharacterTable( "U3(3)" );;  tbl:= CharacterTable( "J4" );;
gap> fus:= InitFusion( subtbl, tbl );;
gap> TestConsistencyMaps( ComputedPowerMaps( subtbl ), fus,
>        ComputedPowerMaps( tbl ) );
true
gap> fus;
[ 1, 2, 4, 4, [ 5, 6 ], [ 5, 6 ], [ 5, 6 ], 10, [ 12, 13 ], [ 12, 13 ], 
  [ 14, 15, 16 ], [ 14, 15, 16 ], [ 21, 22 ], [ 21, 22 ] ]
gap> ConsiderTableAutomorphisms( fus, AutomorphismsOfTable( tbl ) );
[ 9 ]
gap> fus;
[ 1, 2, 4, 4, [ 5, 6 ], [ 5, 6 ], [ 5, 6 ], 10, 12, [ 12, 13 ], 
  [ 14, 15, 16 ], [ 14, 15, 16 ], [ 21, 22 ], [ 21, 22 ] ]
gap> FusionsAllowedByRestrictions( subtbl, tbl, Irr( subtbl ),
>        Irr( tbl ), fus, rec( maxlen:= 10,
>        contained:= ContainedPossibleCharacters, minamb:= 2,
>        maxamb:= infinity, quick:= false ) );
[ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ], 
  [ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 15, 15, 22, 22 ], 
  [ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 16, 16, 22, 22 ] ]

[Top] [Previous] [Up] [Index]