53.3 Operators for Class Functions

chi = psi
chi < psi

Equality and comparison of class functions are defined as for mappings (see Comparisons of Mappings); in case of equal source and range the values components are used to compute the result.

    gap> irr[1]; irr[2];
    Character( S4, [ 1, 1, 1, 1, 1 ] )
    Character( S4, [ 1, 1, 1, -1, -1 ] )
    gap> irr[1] < irr[2];
    false
    gap> irr[1] > irr[2];
    true
    gap> irr[1] = Irr( SolvableGroup( "S4" ) )[1];
    false    # The groups are different. 

chi + psi
chi - psi

+ and - denote the addition and subtraction of class functions.

n * chi
chi * psi

* denotes (besides the composition of mappings, see Operations for Mappings) the multiplication of a class function chi with a scalar n and the tensor product of two class functions.

chi / n

/ denotes the division of the class function chi by a scalar n.

    gap> psi:= irr[3] * irr[4];
    Character( S4, [ 6, -2, 0, 0, 0 ] )
    gap> psi:= irr[3] - irr[1];
    VirtualCharacter( S4, [ 1, 1, -2, -1, -1 ] )
    gap> phi:= psi * irr[4];
    VirtualCharacter( S4, [ 3, -1, 0, -1, 1 ] )
    gap> IsCharacter( phi ); phi;
    true
    Character( S4, [ 3, -1, 0, -1, 1 ] )
    gap> psi:= ( 3 * irr[2] - irr[3] ) * irr[4];
    VirtualCharacter( S4, [ 3, -1, 0, -3, 3 ] )
    gap> 2 * psi ;
    VirtualCharacter( S4, [ 6, -2, 0, -6, 6 ] )
    gap> last / 3;
    ClassFunction( S4, [ 2, -2/3, 0, -2, 2 ] ) 

chi ^ n
g ^ chi

denote the tensor power by a nonnegative integer n and the image of the group element g, like for all mappings (see Operations for Mappings).

chi ^ g

is the conjugate class function by the group element g, that must be an element of the parent of the source of chi or something else that acts on the source via ^. If chi.source is not a permutation group then g may also be a permutation that is interpreted as acting by permuting the classes (This maybe useful for table characters.).

chi ^ G

is the induced class function.

    gap> V4:= Subgroup( S4, S4.generators{ [ 3, 4 ] } );
    Subgroup( S4, [ c, d ] )
    gap> V4.name:= "V4";;
    gap> V4irr:= Irr( V4 );;
    gap> chi:= V4irr[3];
    Character( V4, [ 1, -1, 1, -1 ] )
    gap> chi ^ S4;
    Character( S4, [ 6, -2, 0, 0, 0 ] )
    gap> chi ^ S4.2;
    Character( V4, [ 1, -1, -1, 1 ] )
    gap> chi ^ ( S4.2 ^ 2 );
    Character( V4, [ 1, 1, -1, -1 ] )
    gap> S4.3 ^ chi; S4.4 ^ chi;
    1
    -1
    gap> chi ^ 2;
    Character( V4, [ 1, 1, 1, 1 ] ) 

Previous Up Top Next
Index