68.11 Symmetrizations of Class Functions

Symmetrizations returns the list of symmetrizations of the characters characters of the ordinary character table tbl with the ordinary irreducible characters of the symmetric group of degree n; instead of the integer n, the table of the symmetric group can be entered as Sn.

The symmetrization c^[l] of the character c of tbl with the character l of the symmetric group S_n of degree n is defined by

For special kinds of symmetrizations, see SymmetricParts, AntiSymmetricParts, MinusCharacter and OrthogonalComponents, SymplecticComponents.

Note that the returned list may contain zero class functions, and duplicates are not deleted.

gap> tbl:= CharacterTable( "A5" );;
gap> Symmetrizations( Irr( tbl ){ [ 1 .. 3 ] }, 3 );
[ VirtualCharacter( CharacterTable( "A5" ), [ 0, 0, 0, 0, 0 ] ), 
  VirtualCharacter( CharacterTable( "A5" ), [ 0, 0, 0, 0, 0 ] ), 
  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), 
  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), 
  Character( CharacterTable( "A5" ), [ 8, 0, -1, -E(5)-E(5)^4, -E(5)^2-E(5)^3 
     ] ), Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ), 
  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), 
  Character( CharacterTable( "A5" ), [ 8, 0, -1, -E(5)^2-E(5)^3, -E(5)-E(5)^4 
     ] ), Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ) ]

is the list of symmetrizations of the characters characters of the character table tbl with the trivial character of the symmetric group of degree n (see Symmetrizations).

gap> SymmetricParts( tbl, Irr( tbl ), 3 );
[ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), 
  Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ), 
  Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ), 
  Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ), 
  Character( CharacterTable( "A5" ), [ 35, 3, 2, 0, 0 ] ) ]

is the list of symmetrizations of the characters characters of the character table tbl with the alternating character of the symmetric group of degree n (see Symmetrizations).

gap> AntiSymmetricParts( tbl, Irr( tbl ), 3 );
[ VirtualCharacter( CharacterTable( "A5" ), [ 0, 0, 0, 0, 0 ] ), 
  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), 
  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), 
  Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), 
  Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ) ]

If c is a nonlinear character with indicator +1, a splitting of the tensor power c^m is given by the so-called Murnaghan functions (see Mur58). These components in general have fewer irreducible constituents than the symmetrizations with the symmetric group of degree m (see Symmetrizations).

OrthogonalComponents returns the Murnaghan components of the nonlinear characters of the character table tbl in the list chars up to the power m, where m is an integer between 2 and 6.

The Murnaghan functions are implemented as in Fra82.

Note: If chars is a list of character objects (see IsCharacter) then also the result consists of class function objects. It is not checked whether all characters in chars do really have indicator +1; if there are characters with indicator 0 or -1, the result might contain virtual characters (see also SymplecticComponents), therefore the entries of the result do in general not know that they are characters.

gap> tbl:= CharacterTable( "A8" );;  chi:= Irr( tbl )[2];                      
Character( CharacterTable( "A8" ), [ 7, -1, 3, 4, 1, -1, 1, 2, 0, -1, 0, 0, 
  -1, -1 ] )
gap> OrthogonalComponents( tbl, [ chi ], 3 );
[ ClassFunction( CharacterTable( "A8" ), [ 21, -3, 1, 6, 0, 1, -1, 1, -2, 0, 
      0, 0, 1, 1 ] ), ClassFunction( CharacterTable( "A8" ), 
    [ 27, 3, 7, 9, 0, -1, 1, 2, 1, 0, -1, -1, -1, -1 ] ), 
  ClassFunction( CharacterTable( "A8" ), [ 105, 1, 5, 15, -3, 1, -1, 0, -1, 
      1, 0, 0, 0, 0 ] ), ClassFunction( CharacterTable( "A8" ), 
    [ 35, 3, -5, 5, 2, -1, -1, 0, 1, 0, 0, 0, 0, 0 ] ), 
  ClassFunction( CharacterTable( "A8" ), [ 77, -3, 13, 17, 2, 1, 1, 2, 1, 0, 
      0, 0, 2, 2 ] ) ]

If c is a (nonlinear) character with indicator -1, a splitting of the tensor power c^m is given in terms of the so-called Murnaghan functions (see Mur58). These components in general have fewer irreducible constituents than the symmetrizations with the symmetric group of degree m (see Symmetrizations).

SymplecticComponents returns the symplectic symmetrizations of the nonlinear characters of the character table tbl in the list chars up to the power m, where m is an integer between 2 and 5.

Note: If chars is a list of character objects (see IsCharacter) then also the result consists of class function objects. It is not checked whether all characters in chars do really have indicator -1; if there are characters with indicator 0 or +1, the result might contain virtual characters (see also OrthogonalComponents), therefore the entries of the result do in general not know that they are characters.

gap> tbl:= CharacterTable( "U3(3)" );;  chi:= Irr( tbl )[2];                   
Character( CharacterTable( "U3(3)" ), [ 6, -2, -3, 0, -2, -2, 2, 1, -1, -1, 
  0, 0, 1, 1 ] )
gap> SymplecticComponents( tbl, [ chi ], 3 );                                 
[ ClassFunction( CharacterTable( "U3(3)" ), [ 14, -2, 5, -1, 2, 2, 2, 1, 0, 
      0, 0, 0, -1, -1 ] ), ClassFunction( CharacterTable( "U3(3)" ), 
    [ 21, 5, 3, 0, 1, 1, 1, -1, 0, 0, -1, -1, 1, 1 ] ), 
  ClassFunction( CharacterTable( "U3(3)" ), [ 64, 0, -8, -2, 0, 0, 0, 0, 1, 
      1, 0, 0, 0, 0 ] ), ClassFunction( CharacterTable( "U3(3)" ), 
    [ 14, 6, -4, 2, -2, -2, 2, 0, 0, 0, 0, 0, -2, -2 ] ), 
  ClassFunction( CharacterTable( "U3(3)" ), [ 56, -8, 2, 2, 0, 0, 0, -2, 0, 
      0, 0, 0, 0, 0 ] ) ]

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