  
  [1X4 Idempotents[0X
  
  
  [1X4.1 Computing idempotents from character table[0X
  
  [1X4.1-1 PrimitiveCentralIdempotentsByCharacterTable[0m
  
  [2X> PrimitiveCentralIdempotentsByCharacterTable( [0X[3XFG[0X[2X ) _______________[0Xoperation
  [6XReturns:[0X  A list of group algebra elements.
  
  The input [3XFG[0m should be a semisimple group algebra.
  
  Returns  the list of primitive central idempotents of [3XFG[0m using the character
  table of G ([14X7.4[0m).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> QS3 := GroupRing( Rationals, SymmetricGroup(3) );;                 [0X
    [4Xgap> PrimitiveCentralIdempotentsByCharacterTable( QS3 );[0X
    [4X[ (1/6)*()+(-1/6)*(2,3)+(-1/6)*(1,2)+(1/6)*(1,2,3)+(1/6)*(1,3,2)+(-1/6)*(1,3),[0X
    [4X  (2/3)*()+(-1/3)*(1,2,3)+(-1/3)*(1,3,2), (1/6)*()+(1/6)*(2,3)+(1/6)*(1,2)+(1/[0X
    [4X    6)*(1,2,3)+(1/6)*(1,3,2)+(1/6)*(1,3) ][0X
    [4Xgap> QG:=GroupRing( Rationals , SmallGroup(24,3) );[0X
    [4X<algebra-with-one over Rationals, with 4 generators>[0X
    [4Xgap> FG:=GroupRing( CF(3) , SmallGroup(24,3) );[0X
    [4X<algebra-with-one over CF(3), with 4 generators>[0X
    [4Xgap> pciQG := PrimitiveCentralIdempotentsByCharacterTable(QG);;[0X
    [4Xgap> pciFG := PrimitiveCentralIdempotentsByCharacterTable(FG);;[0X
    [4Xgap> Length(pciQG);[0X
    [4X5[0X
    [4Xgap> Length(pciFG);[0X
    [4X7[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.2 Testing lists of idempotents for completeness[0X
  
  [1X4.2-1 IsCompleteSetOfOrthogonalIdempotents[0m
  
  [2X> IsCompleteSetOfOrthogonalIdempotents( [0X[3XR, list[0X[2X ) _________________[0Xoperation
  
  The input should be formed by a unital ring [3XR[0m and a list [3Xlist[0m of elements of
  [3XR[0m.
  
  Returns  [9Xtrue[0m  if the list [3Xlist[0m is a complete list of orthogonal idempotents
  of  [3XR[0m.  That  is,  the  output is [9Xtrue[0m provided the following conditions are
  satisfied:
  
  * The sum of the elements of [3Xlist[0m is the identity of [3XR[0m,
  
  * e^2=e, for every e in [3Xlist[0m and
  
  * e*f=0, if e and f are elements in different positions of [3Xlist[0m.
  
  No claim is made on the idempotents being central or primitive.
  
  Note  that  the  if  a  non-zero  element  t  of  [3XR[0m appears in two different
  positions  of [3Xlist[0m then the output is [9Xfalse[0m, and that the list [3Xlist[0m must not
  contain zeroes.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> QS5 := GroupRing( Rationals, SymmetricGroup(5) );;[0X
    [4Xgap> idemp := PrimitiveCentralIdempotentsByCharacterTable( QS5 );;[0X
    [4Xgap> IsCompleteSetOfOrthogonalIdempotents( QS5, idemp );[0X
    [4Xtrue[0X
    [4Xgap> IsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ) ] );[0X
    [4Xtrue[0X
    [4Xgap> IsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ), One( QS5 ) ] );[0X
    [4Xfalse[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.3 Idempotents from Shoda pairs[0X
  
  [1X4.3-1 PrimitiveCentralIdempotentsByStrongSP[0m
  
  [2X> PrimitiveCentralIdempotentsByStrongSP( [0X[3XFG[0X[2X ) _____________________[0Xattribute
  [6XReturns:[0X  A list of group algebra elements.
  
  The  input [3XFG[0m 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 [3XFG[0m realizable by strong Shoda pairs ([14X7.15[0m) of G.
  
  If  F  is  a  finite  field then the output is the list of primitive central
  idempotents  of  [3XFG[0m  realizable  by  strong  Shoda  pairs  (K,H)  of  G  and
  q-cyclotomic classes modulo the index of H in K ([14X7.17[0m).
  
  If  the  list  of  primitive  central idempotents given by the output is not
  complete  (i.e.  if  the  group  G  is  not [13Xstrongly monomial[0m ([14X7.16[0m)) then a
  warning is displayed.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> QG:=GroupRing( Rationals, AlternatingGroup(4) );;           [0X
    [4Xgap> PrimitiveCentralIdempotentsByStrongSP( QG );[0X
    [4X[ (1/12)*()+(1/12)*(2,3,4)+(1/12)*(2,4,3)+(1/12)*(1,2)(3,4)+(1/12)*(1,2,3)+(1/[0X
    [4X    12)*(1,2,4)+(1/12)*(1,3,2)+(1/12)*(1,3,4)+(1/12)*(1,3)(2,4)+(1/12)*[0X
    [4X    (1,4,2)+(1/12)*(1,4,3)+(1/12)*(1,4)(2,3),[0X
    [4X  (1/6)*()+(-1/12)*(2,3,4)+(-1/12)*(2,4,3)+(1/6)*(1,2)(3,4)+(-1/12)*(1,2,3)+([0X
    [4X    -1/12)*(1,2,4)+(-1/12)*(1,3,2)+(-1/12)*(1,3,4)+(1/6)*(1,3)(2,4)+(-1/12)*[0X
    [4X    (1,4,2)+(-1/12)*(1,4,3)+(1/6)*(1,4)(2,3),[0X
    [4X  (3/4)*()+(-1/4)*(1,2)(3,4)+(-1/4)*(1,3)(2,4)+(-1/4)*(1,4)(2,3) ][0X
    [4Xgap> QG := GroupRing( Rationals, SmallGroup(24,3) );;[0X
    [4Xgap> PrimitiveCentralIdempotentsByStrongSP( QG );;[0X
    [4XWedderga: Warning!!![0X
    [4XThe output is a NON-COMPLETE list of prim. central idemp.s of the input! [0X
    [4Xgap> FG := GroupRing( GF(2), Group((1,2,3)) );;[0X
    [4Xgap> PrimitiveCentralIdempotentsByStrongSP( FG );[0X
    [4X[ (Z(2)^0)*()+(Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2), [0X
    [4X  (Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2) ][0X
    [4Xgap> FG := GroupRing( GF(5), SmallGroup(24,3) );; [0X
    [4Xgap> PrimitiveCentralIdempotentsByStrongSP( FG );;[0X
    [4XWedderga: Warning!!![0X
    [4XThe output is a NON-COMPLETE list of prim. central idemp.s of the input! [0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.3-2 PrimitiveCentralIdempotentsBySP[0m
  
  [2X> PrimitiveCentralIdempotentsBySP( [0X[3XQG[0X[2X ) ____________________________[0Xfunction
  [6XReturns:[0X  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 [3XQG[0m 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 [3XQG[0m if and
  only if G is monomial, otherwise a warning message is displayed.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> QG := GroupRing( Rationals, SymmetricGroup(4) );[0X
    [4X<algebra-with-one over Rationals, with 2 generators>[0X
    [4Xgap> pci:=PrimitiveCentralIdempotentsBySP( QG );[0X
    [4X[ (1/24)*()+(1/24)*(3,4)+(1/24)*(2,3)+(1/24)*(2,3,4)+(1/24)*(2,4,3)+(1/24)*[0X
    [4X    (2,4)+(1/24)*(1,2)+(1/24)*(1,2)(3,4)+(1/24)*(1,2,3)+(1/24)*(1,2,3,4)+(1/[0X
    [4X    24)*(1,2,4,3)+(1/24)*(1,2,4)+(1/24)*(1,3,2)+(1/24)*(1,3,4,2)+(1/24)*[0X
    [4X    (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)+([0X
    [4X    1/24)*(1,4,2)+(1/24)*(1,4,3)+(1/24)*(1,4)+(1/24)*(1,4,2,3)+(1/24)*(1,4)[0X
    [4X    (2,3), (1/24)*()+(-1/24)*(3,4)+(-1/24)*(2,3)+(1/24)*(2,3,4)+(1/24)*[0X
    [4X    (2,4,3)+(-1/24)*(2,4)+(-1/24)*(1,2)+(1/24)*(1,2)(3,4)+(1/24)*(1,2,3)+(-1/[0X
    [4X    24)*(1,2,3,4)+(-1/24)*(1,2,4,3)+(1/24)*(1,2,4)+(1/24)*(1,3,2)+(-1/24)*[0X
    [4X    (1,3,4,2)+(-1/24)*(1,3)+(1/24)*(1,3,4)+(1/24)*(1,3)(2,4)+(-1/24)*[0X
    [4X    (1,3,2,4)+(-1/24)*(1,4,3,2)+(1/24)*(1,4,2)+(1/24)*(1,4,3)+(-1/24)*(1,4)+([0X
    [4X    -1/24)*(1,4,2,3)+(1/24)*(1,4)(2,3), (3/8)*()+(-1/8)*(3,4)+(-1/8)*(2,3)+([0X
    [4X    -1/8)*(2,4)+(-1/8)*(1,2)+(-1/8)*(1,2)(3,4)+(1/8)*(1,2,3,4)+(1/8)*[0X
    [4X    (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)+([0X
    [4X    1/8)*(1,4,3,2)+(-1/8)*(1,4)+(1/8)*(1,4,2,3)+(-1/8)*(1,4)(2,3), [0X
    [4X  (3/8)*()+(1/8)*(3,4)+(1/8)*(2,3)+(1/8)*(2,4)+(1/8)*(1,2)+(-1/8)*(1,2)(3,4)+([0X
    [4X    -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)[0X
    [4X    (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/[0X
    [4X    8)*(1,4)(2,3), (1/6)*()+(-1/12)*(2,3,4)+(-1/12)*(2,4,3)+(1/6)*(1,2)(3,4)+([0X
    [4X    -1/12)*(1,2,3)+(-1/12)*(1,2,4)+(-1/12)*(1,3,2)+(-1/12)*(1,3,4)+(1/6)*(1,3)[0X
    [4X    (2,4)+(-1/12)*(1,4,2)+(-1/12)*(1,4,3)+(1/6)*(1,4)(2,3) ][0X
    [4Xgap> IsCompleteSetOfPCIs(QG,pci);[0X
    [4Xtrue[0X
    [4Xgap> QS5 := GroupRing( Rationals, SymmetricGroup(5) );;[0X
    [4Xgap> pci:=PrimitiveCentralIdempotentsBySP( QS5 );;[0X
    [4XWedderga: Warning!![0X
    [4XThe output is a NON-COMPLETE list of prim. central idemp.s of the input![0X
    [4Xgap> IsCompleteSetOfPCIs( QS5 , pci );[0X
    [4Xfalse[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  The   output  of  [2XPrimitiveCentralIdempotentsBySP[0m  contains  the  output  of
  [2XPrimitiveCentralIdempotentsByStrongSP[0m ([14X4.3-1[0m), possibly properly.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> QG := GroupRing( Rationals, SmallGroup(48,28) );;[0X
    [4Xgap> pci:=PrimitiveCentralIdempotentsBySP( QG );;[0X
    [4XWedderga: Warning!![0X
    [4XThe output is a NON-COMPLETE list of prim. central idemp.s of the input! [0X
    [4Xgap> Length(pci);    [0X
    [4X6[0X
    [4Xgap> spci:=PrimitiveCentralIdempotentsByStrongSP( QG );;  [0X
    [4XWedderga: Warning!!![0X
    [4XThe output is a NON-COMPLETE list of prim. central idemp.s of the input! [0X
    [4Xgap> Length(spci);[0X
    [4X5[0X
    [4Xgap> IsSubset(pci,spci);          [0X
    [4Xtrue[0X
    [4Xgap> QG:=GroupRing(Rationals,SmallGroup(1000,86));[0X
    [4X<algebra-with-one over Rationals, with 6 generators>[0X
    [4Xgap> IsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsBySP(QG) );[0X
    [4Xtrue[0X
    [4Xgap> IsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsByStrongSP(QG) );[0X
    [4XWedderga: Warning!!![0X
    [4XThe output is a NON-COMPLETE list of prim. central idemp.s of the input![0X
    [4Xfalse[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
