68.7 Creating Class Functions using Groups

is the trivial character of the group G or its character table tbl, respectively. This is the class function with value equal to 1 for each class.

gap> TrivialCharacter( CharacterTable( "A5" ) );
Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] )
gap> TrivialCharacter( SymmetricGroup( 3 ) );
Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, 1, 1 ] )

If the argument is a permutation group G then NaturalCharacter returns the (ordinary) character of the natural permutation representation of G on the set of moved points (see MovedPoints), that is, the value on each class is the number of points among the moved points of G that are fixed by any permutation in that class.

If the argument is a matrix group G in characteristic zero then NaturalCharacter returns the (ordinary) character of the natural matrix representation of G, that is, the value on each class is the trace of any matrix in that class.

If the argument is a group homomorphism hom whose image is a permutation group or a matrix group then NaturalCharacter returns the restriction of the natural character of the image of hom to the preimage of hom.

gap> NaturalCharacter( SymmetricGroup( 3 ) );
Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 3, 1, 0 ] )
gap> NaturalCharacter( Group( [ [ 0, -1 ], [ 1, -1 ] ] ) );
Character( CharacterTable( Group([ [ [ 0, -1 ], [ 1, -1 ] ] ]) ), 
[ 2, -1, -1 ] )
gap> d8:= DihedralGroup( 8 );;  hom:= IsomorphismPermGroup( d8 );;
gap> NaturalCharacter( hom );
Character( CharacterTable( <pc group of size 8 with 3 generators> ), 
[ 8, 0, 0, 0, 0 ] )

Called with a group G, an action domain or proper set D, and an action function opr (see Chapter Group Actions), PermutationCharacter returns the permutation character of the action of G on D via opr, that is, the value on each class is the number of points in D that are fixed by an element in this class under the action opr.

If the arguments are a group G and a subgroup U of G then PermutationCharacter returns the permutation character of the action of G on the right cosets of U via right multiplication.

To compute the permutation character of a transitive permutation group G on the cosets of a point stabilizer U, the attribute NaturalCharacter( G ) can be used instead of PermutationCharacter( G, U ).

More facilities concerning permutation characters are the transitivity test (see Section Operations for Class Functions) and several tools for computing possible permutation characters (see Possible Permutation Characters, Computing Possible Permutation Characters).

gap> PermutationCharacter( GL(2,2), AsSSortedList( GF(2)^2 ), OnRight );
Character( CharacterTable( SL(2,2) ), [ 4, 2, 1 ] )
gap> s3:= SymmetricGroup( 3 );;  a3:= DerivedSubgroup( s3 );;
gap> PermutationCharacter( s3, a3 );
Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 2, 0, 2 ] )

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