4 Idempotents
  
  
  4.1 Computing idempotents from character table
  
  4.1-1 PrimitiveCentralIdempotentsByCharacterTable
  
  > PrimitiveCentralIdempotentsByCharacterTable( FG ) _______________operation
  Returns:  A list of group algebra elements.
  
  The input FG should be a semisimple group algebra.
  
  Returns  the list of primitive central idempotents of FG using the character
  table of G (7.4).
  
  ---------------------------  Example  ----------------------------
    
    gap> QS3 := GroupRing( Rationals, SymmetricGroup(3) );;                 
    gap> PrimitiveCentralIdempotentsByCharacterTable( QS3 );
    [ (1/6)*()+(-1/6)*(2,3)+(-1/6)*(1,2)+(1/6)*(1,2,3)+(1/6)*(1,3,2)+(-1/6)*(1,3),
      (2/3)*()+(-1/3)*(1,2,3)+(-1/3)*(1,3,2), (1/6)*()+(1/6)*(2,3)+(1/6)*(1,2)+(1/
        6)*(1,2,3)+(1/6)*(1,3,2)+(1/6)*(1,3) ]
    gap> QG:=GroupRing( Rationals , SmallGroup(24,3) );
    <algebra-with-one over Rationals, with 4 generators>
    gap> FG:=GroupRing( CF(3) , SmallGroup(24,3) );
    <algebra-with-one over CF(3), with 4 generators>
    gap> pciQG := PrimitiveCentralIdempotentsByCharacterTable(QG);;
    gap> pciFG := PrimitiveCentralIdempotentsByCharacterTable(FG);;
    gap> Length(pciQG);
    5
    gap> Length(pciFG);
    7
    
  ------------------------------------------------------------------
  
  
  4.2 Testing lists of idempotents for completeness
  
  4.2-1 IsCompleteSetOfOrthogonalIdempotents
  
  > IsCompleteSetOfOrthogonalIdempotents( R, list ) _________________operation
  
  The input should be formed by a unital ring R and a list list of elements of
  R.
  
  Returns  true  if the list list is a complete list of orthogonal idempotents
  of  R.  That  is,  the  output is true provided the following conditions are
  satisfied:
  
  * The sum of the elements of list is the identity of R,
  
  * e^2=e, for every e in list and
  
  * e*f=0, if e and f are elements in different positions of list.
  
  No claim is made on the idempotents being central or primitive.
  
  Note  that  the  if  a  non-zero  element  t  of  R appears in two different
  positions  of list then the output is false, and that the list list must not
  contain zeroes.
  
  ---------------------------  Example  ----------------------------
    
    gap> QS5 := GroupRing( Rationals, SymmetricGroup(5) );;
    gap> idemp := PrimitiveCentralIdempotentsByCharacterTable( QS5 );;
    gap> IsCompleteSetOfOrthogonalIdempotents( QS5, idemp );
    true
    gap> IsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ) ] );
    true
    gap> IsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ), One( QS5 ) ] );
    false
    
  ------------------------------------------------------------------
  
  
  4.3 Idempotents from Shoda pairs
  
  4.3-1 PrimitiveCentralIdempotentsByStrongSP
  
  > PrimitiveCentralIdempotentsByStrongSP( FG ) _____________________attribute
  Returns:  A list of group algebra elements.
  
  The  input FG should be a semisimple group algebra of a finite group G whose
  coefficient field F is either a finite field or the field ℚ of rationals.
  
  If F = ℚ then the output is the list of primitive central idempotents of the
  group algebra FG realizable by strong Shoda pairs (7.15) of G.
  
  If  F  is  a  finite  field then the output is the list of primitive central
  idempotents  of  FG  realizable  by  strong  Shoda  pairs  (K,H)  of  G  and
  q-cyclotomic classes modulo the index of H in K (7.17).
  
  If  the  list  of  primitive  central idempotents given by the output is not
  complete  (i.e.  if  the  group  G  is  not strongly monomial (7.16)) then a
  warning is displayed.
  
  ---------------------------  Example  ----------------------------
    
    gap> QG:=GroupRing( Rationals, AlternatingGroup(4) );;           
    gap> PrimitiveCentralIdempotentsByStrongSP( QG );
    [ (1/12)*()+(1/12)*(2,3,4)+(1/12)*(2,4,3)+(1/12)*(1,2)(3,4)+(1/12)*(1,2,3)+(1/
        12)*(1,2,4)+(1/12)*(1,3,2)+(1/12)*(1,3,4)+(1/12)*(1,3)(2,4)+(1/12)*
        (1,4,2)+(1/12)*(1,4,3)+(1/12)*(1,4)(2,3),
      (1/6)*()+(-1/12)*(2,3,4)+(-1/12)*(2,4,3)+(1/6)*(1,2)(3,4)+(-1/12)*(1,2,3)+(
        -1/12)*(1,2,4)+(-1/12)*(1,3,2)+(-1/12)*(1,3,4)+(1/6)*(1,3)(2,4)+(-1/12)*
        (1,4,2)+(-1/12)*(1,4,3)+(1/6)*(1,4)(2,3),
      (3/4)*()+(-1/4)*(1,2)(3,4)+(-1/4)*(1,3)(2,4)+(-1/4)*(1,4)(2,3) ]
    gap> QG := GroupRing( Rationals, SmallGroup(24,3) );;
    gap> PrimitiveCentralIdempotentsByStrongSP( QG );;
    Wedderga: Warning!!!
    The output is a NON-COMPLETE list of prim. central idemp.s of the input! 
    gap> FG := GroupRing( GF(2), Group((1,2,3)) );;
    gap> PrimitiveCentralIdempotentsByStrongSP( FG );
    [ (Z(2)^0)*()+(Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2), 
      (Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2) ]
    gap> FG := GroupRing( GF(5), SmallGroup(24,3) );; 
    gap> PrimitiveCentralIdempotentsByStrongSP( FG );;
    Wedderga: Warning!!!
    The output is a NON-COMPLETE list of prim. central idemp.s of the input! 
    
  ------------------------------------------------------------------
  
  4.3-2 PrimitiveCentralIdempotentsBySP
  
  > PrimitiveCentralIdempotentsBySP( QG ) ____________________________function
  Returns:  A list of group algebra elements.
  
  The input should be a rational group algebra of a finite group G.
  
  Returns  a  list  containing  all the primitive central idempotents e of the
  rational group algebra QG such that chi(e)ne 0 for some irreducible monomial
  character chi of G.
  
  The  output  is  the  list of all primitive central idempotents of QG if and
  only if G is monomial, otherwise a warning message is displayed.
  
  ---------------------------  Example  ----------------------------
    
    gap> QG := GroupRing( Rationals, SymmetricGroup(4) );
    <algebra-with-one over Rationals, with 2 generators>
    gap> pci:=PrimitiveCentralIdempotentsBySP( QG );
    [ (1/24)*()+(1/24)*(3,4)+(1/24)*(2,3)+(1/24)*(2,3,4)+(1/24)*(2,4,3)+(1/24)*
        (2,4)+(1/24)*(1,2)+(1/24)*(1,2)(3,4)+(1/24)*(1,2,3)+(1/24)*(1,2,3,4)+(1/
        24)*(1,2,4,3)+(1/24)*(1,2,4)+(1/24)*(1,3,2)+(1/24)*(1,3,4,2)+(1/24)*
        (1,3)+(1/24)*(1,3,4)+(1/24)*(1,3)(2,4)+(1/24)*(1,3,2,4)+(1/24)*(1,4,3,2)+(
        1/24)*(1,4,2)+(1/24)*(1,4,3)+(1/24)*(1,4)+(1/24)*(1,4,2,3)+(1/24)*(1,4)
        (2,3), (1/24)*()+(-1/24)*(3,4)+(-1/24)*(2,3)+(1/24)*(2,3,4)+(1/24)*
        (2,4,3)+(-1/24)*(2,4)+(-1/24)*(1,2)+(1/24)*(1,2)(3,4)+(1/24)*(1,2,3)+(-1/
        24)*(1,2,3,4)+(-1/24)*(1,2,4,3)+(1/24)*(1,2,4)+(1/24)*(1,3,2)+(-1/24)*
        (1,3,4,2)+(-1/24)*(1,3)+(1/24)*(1,3,4)+(1/24)*(1,3)(2,4)+(-1/24)*
        (1,3,2,4)+(-1/24)*(1,4,3,2)+(1/24)*(1,4,2)+(1/24)*(1,4,3)+(-1/24)*(1,4)+(
        -1/24)*(1,4,2,3)+(1/24)*(1,4)(2,3), (3/8)*()+(-1/8)*(3,4)+(-1/8)*(2,3)+(
        -1/8)*(2,4)+(-1/8)*(1,2)+(-1/8)*(1,2)(3,4)+(1/8)*(1,2,3,4)+(1/8)*
        (1,2,4,3)+(1/8)*(1,3,4,2)+(-1/8)*(1,3)+(-1/8)*(1,3)(2,4)+(1/8)*(1,3,2,4)+(
        1/8)*(1,4,3,2)+(-1/8)*(1,4)+(1/8)*(1,4,2,3)+(-1/8)*(1,4)(2,3), 
      (3/8)*()+(1/8)*(3,4)+(1/8)*(2,3)+(1/8)*(2,4)+(1/8)*(1,2)+(-1/8)*(1,2)(3,4)+(
        -1/8)*(1,2,3,4)+(-1/8)*(1,2,4,3)+(-1/8)*(1,3,4,2)+(1/8)*(1,3)+(-1/8)*(1,3)
        (2,4)+(-1/8)*(1,3,2,4)+(-1/8)*(1,4,3,2)+(1/8)*(1,4)+(-1/8)*(1,4,2,3)+(-1/
        8)*(1,4)(2,3), (1/6)*()+(-1/12)*(2,3,4)+(-1/12)*(2,4,3)+(1/6)*(1,2)(3,4)+(
        -1/12)*(1,2,3)+(-1/12)*(1,2,4)+(-1/12)*(1,3,2)+(-1/12)*(1,3,4)+(1/6)*(1,3)
        (2,4)+(-1/12)*(1,4,2)+(-1/12)*(1,4,3)+(1/6)*(1,4)(2,3) ]
    gap> IsCompleteSetOfPCIs(QG,pci);
    true
    gap> QS5 := GroupRing( Rationals, SymmetricGroup(5) );;
    gap> pci:=PrimitiveCentralIdempotentsBySP( QS5 );;
    Wedderga: Warning!!
    The output is a NON-COMPLETE list of prim. central idemp.s of the input!
    gap> IsCompleteSetOfPCIs( QS5 , pci );
    false
    
  ------------------------------------------------------------------
  
  The   output  of  PrimitiveCentralIdempotentsBySP  contains  the  output  of
  PrimitiveCentralIdempotentsByStrongSP (4.3-1), possibly properly.
  
  ---------------------------  Example  ----------------------------
    
    gap> QG := GroupRing( Rationals, SmallGroup(48,28) );;
    gap> pci:=PrimitiveCentralIdempotentsBySP( QG );;
    Wedderga: Warning!!
    The output is a NON-COMPLETE list of prim. central idemp.s of the input! 
    gap> Length(pci);    
    6
    gap> spci:=PrimitiveCentralIdempotentsByStrongSP( QG );;  
    Wedderga: Warning!!!
    The output is a NON-COMPLETE list of prim. central idemp.s of the input! 
    gap> Length(spci);
    5
    gap> IsSubset(pci,spci);          
    true
    gap> QG:=GroupRing(Rationals,SmallGroup(1000,86));
    <algebra-with-one over Rationals, with 6 generators>
    gap> IsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsBySP(QG) );
    true
    gap> IsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsByStrongSP(QG) );
    Wedderga: Warning!!!
    The output is a NON-COMPLETE list of prim. central idemp.s of the input!
    false
    
  ------------------------------------------------------------------