For restricting a class function of a group G to a subgroup H and for inducing a class function of H to G, the class fusion from H to G must be known (see Class Fusions between Character Tables).
If F is the factor group of G by the normal subgroup N then each class function of F can be naturally regarded as a class function of G, with N in its kernel. For a class function of F, the corresponding class function of G is called the inflated class function. Restriction and inflation are in principle the same, namely indirection of a class function by the appropriate fusion map, and thus no extra operation is needed for this process. But note that contrary to the case of a subgroup fusion, the factor fusion can in general not be computed from the groups G and F; either one needs the natural homomorphism or the factor fusion to the character table of F must be stored on the table of G. This explains the different syntax for computing restricted and inflated class functions.
In the following, the meaning of the optional first argument tbl is the same as in Section Operations for Class Functions.
RestrictedClassFunction( [tbl, ]chi, H ) O
RestrictedClassFunction( [tbl, ]chi, hom ) O
RestrictedClassFunction( [tbl, ]chi, subtbl ) O
For a class function chi of the group G, say,
and either a subgroup H of G
or a homomorphism from H to G
or the character table subtbl of this subgroup,
RestrictedClassFunction returns the class function of H obtained by
restricting chi to H.
In the situation that chi is a class function of a factor group F of H, the variant where hom is a homomorphism can be always used, the calls with argument H or subtbl work only if the factor fusion is stored on the character table.
RestrictedClassFunctions( [tbl, ]chars, H ) O
RestrictedClassFunctions( [tbl, ]chars, hom ) O
RestrictedClassFunctions( [tbl, ]chars, subtbl ) O
RestrictedClassFunctions is similar to RestrictedClassFunction
(see RestrictedClassFunction),
the only difference is that it takes a list chars of class functions
instead of one class function, and returns the list of restricted class
functions.
gap> a5:= CharacterTable( "A5" );; s5:= CharacterTable( "S5" );; gap> RestrictedClassFunction( Irr( s5 )[2], a5 ); Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ) gap> RestrictedClassFunctions( Irr( s5 ), a5 ); [ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 6, -2, 0, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ) ] gap> hom:= NaturalHomomorphismByNormalSubgroup( S4, der );; gap> RestrictedClassFunctions( Irr( Image( hom ) ), hom ); [ Character( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( S4 ), [ 1, -1, 1, 1, -1 ] ) ]
InducedClassFunction( [tbl, ]chi, H ) O
InducedClassFunction( [tbl, ]chi, hom ) O
InducedClassFunction( [tbl, ]chi, suptbl ) O
For a class function chi of the group G, say,
and either a supergroup H of G
or a homomorphism from G to H
or the character table suptbl of this supergroup,
InducedClassFunction returns the class function of H obtained by
inducing chi to H.
InducedClassFunctions( [tbl, ]chars, H ) O
InducedClassFunctions( [tbl, ]chars, hom ) O
InducedClassFunctions( [tbl, ]chars, suptbl ) O
InducedClassFunctions is similar to InducedClassFunction
(see InducedClassFunction),
the only difference is that it takes a list chars of class functions
instead of one class function, and returns the list of induced class
functions.
gap> InducedClassFunctions( Irr( a5 ), s5 ); [ Character( CharacterTable( "A5.2" ), [ 2, 2, 2, 2, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 6, -2, 0, 1, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 6, -2, 0, 1, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 8, 0, 2, -2, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 10, 2, -2, 0, 0, 0, 0 ] ) ]
InducedCyclic( tbl ) O
InducedCyclic( tbl, "all" ) O
InducedCyclic( tbl, classes ) O
InducedCyclic( tbl, classes, "all" ) O
InducedCyclic calculates characters induced up from cyclic subgroups
of the ordinary character table tbl to tbl,
and returns the strictly sorted list of the induced characters.
If "all" is specified then all irreducible characters of these
subgroups are induced,
otherwise only the permutation characters are calculated.
If a list classes is specified then only those cyclic subgroups generated by these classes are considered, otherwise all classes of tbl are considered.
gap> InducedCyclic( a5, "all" );
[ Character( CharacterTable( "A5" ), [ 12, 0, 0, 2, 2 ] ),
Character( CharacterTable( "A5" ), [ 12, 0, 0, E(5)^2+E(5)^3, E(5)+E(5)^4
] ), Character( CharacterTable( "A5" ),
[ 12, 0, 0, E(5)+E(5)^4, E(5)^2+E(5)^3 ] ),
Character( CharacterTable( "A5" ), [ 20, 0, -1, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 30, -2, 0, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 30, 2, 0, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 60, 0, 0, 0, 0 ] ) ]
[Top] [Previous] [Up] [Next] [Index]
GAP 4 manual