54.5 TestHomogeneous

TestHomogeneous( chi, N )

returns a record with information whether the restriction of the character chi of the group G to the normal subgroup N of G is homogeneous, i.e., is a multiple of an irreducible character of N.

N may be given also as list of conjugacy class positions w.r. to G.

The components of the result are

isHomogeneous:

true or false,

comment:

a string telling a reason for the value of the isHomogeneous component,

character:

irreducible constituent of the restriction, only bound if the restriction had to be checked,

multiplicity:

multiplicity of the character component in the restriction of chi.

    gap> chi:= Irr( Sl23 )[4];
    Character( Sl(2,3), [ 2, -2, 0, -1, 1, -1, 1 ] )
    gap> n:= NormalSubgroupClasses( Sl23, [ 1, 2, 3 ] );
    Subgroup( Sl(2,3), [ b, c, d ] )
    gap> TestHomogeneous( chi, [ 1, 2, 3 ] );
    rec(
      isHomogeneous := true,
      comment := "restricts irreducibly" )
    gap> chi:= Irr( Sl23 )[7];
    Character( Sl(2,3), [ 3, 3, -1, 0, 0, 0, 0 ] )
    gap> TestHomogeneous( chi, n );
    #W  Warning: Group has no name
    rec(
      isHomogeneous := false,
      comment := "restriction checked",
      character := Character( Subgroup( Sl(2,3), [ b, c, d ] ),
        [ 1, 1, -1, 1, -1 ] ),
      multiplicity := 1 ) 

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