68.6 Creating Class Functions from Values Lists

In the first form, ClassFunction returns the class function of the character table tbl with values given by the list values of cyclotomics. In the second form, G must be a group, and the class function of its ordinary character table is returned.

Note that tbl determines the underying characteristic of the returned class function (see UnderlyingCharacteristic).

VirtualCharacter returns the virtual character (see IsVirtualCharacter) of the character table tbl or the group G, respectively, with values given by the list values.

It is not checked whether the given values really describe a virtual character.

Character returns the character (see IsCharacter) of the character table tbl or the group G, respectively, with values given by the list values.

It is not checked whether the given values really describe a character.

gap> g:= DihedralGroup( 8 );  tbl:= CharacterTable( g );
<pc group of size 8 with 3 generators>
CharacterTable( <pc group of size 8 with 3 generators> )
gap> SetName( tbl, "D8" );
gap> phi:= ClassFunction( g, [ 1, -1, 0, 2, -2 ] );
ClassFunction( D8, [ 1, -1, 0, 2, -2 ] )
gap> psi:= ClassFunction( tbl,
>              List( Irr( g ), chi -> ScalarProduct( chi, phi ) ) );
ClassFunction( D8, [ -3/8, 5/8, 9/8, 1/8, -1/4 ] )
gap> chi:= VirtualCharacter( g, [ 0, 0, 8, 0, 0 ] );
VirtualCharacter( D8, [ 0, 0, 8, 0, 0 ] )
gap> reg:= Character( tbl, [ 8, 0, 0, 0, 0 ] );
Character( D8, [ 8, 0, 0, 0, 0 ] )

Let tbl be a character table, chi a class function object (not necessarily a class function of tbl), and values a list of cyclotomics. ClassFunctionSameType returns the class function y of tbl with values list values, constructed with ClassFunction (see ClassFunction).

If chi is known to be a (virtual) character then y is also known to be a (virtual) character.

gap> h:= Centre( g );;
gap> centbl:= CharacterTable( h );;  SetName( centbl, "C2" );
gap> ClassFunctionSameType( centbl, phi, [ 1, 1 ] );
ClassFunction( C2, [ 1, 1 ] )
gap> ClassFunctionSameType( centbl, chi, [ 1, 1 ] );
VirtualCharacter( C2, [ 1, 1 ] )
gap> ClassFunctionSameType( centbl, reg, [ 1, 1 ] );
Character( C2, [ 1, 1 ] )

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