> 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).
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
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> 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.
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 |
> 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.
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!
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> 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.
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
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The output of PrimitiveCentralIdempotentsBySP contains the output of PrimitiveCentralIdempotentsByStrongSP (4.3-1), possibly properly.
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 |
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