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### 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 $$G$$ (9.4).


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



#### 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:

$$\cdot$$ The sum of the elements of list is the identity of R,

$$\cdot$$ $$e^2=e$$, for every $$e$$ in list and

$$\cdot$$ $$e*f=0$$, 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



#### 4.3 Idempotents from Shoda pairs

##### 4.3-1 PrimitiveCentralIdempotentsByESSP
 ‣ PrimitiveCentralIdempotentsByESSP( QG ) ( attribute )

Returns: A list of group algebra elements.

The input QG should be a semisimple rational group algebra of a finite group $$G$$.

The output is the list of primitive central idempotents of the group algebra QG realizable by extremely strong Shoda pairs (9.16) of $$G$$.

If the list of primitive central idempotents given by the output is not complete (i.e. if the group $$G$$ is not normally monomial (9.18)) then a warning is displayed.


gap> QG:=GroupRing( Rationals, DihedralGroup(16) );;
gap> PrimitiveCentralIdempotentsByESSP( QG );
[ (1/16)*<identity> of ...+(1/16)*f1+(1/16)*f2+(1/16)*f3+(1/16)*f4+(1/
16)*f1*f2+(1/16)*f1*f3+(1/16)*f1*f4+(1/16)*f2*f3+(1/16)*f2*f4+(1/
16)*f3*f4+(1/16)*f1*f2*f3+(1/16)*f1*f2*f4+(1/16)*f1*f3*f4+(1/
16)*f2*f3*f4+(1/16)*f1*f2*f3*f4, (1/16)*<identity> of ...+(-1/16)*f1+(-1/
16)*f2+(1/16)*f3+(1/16)*f4+(1/16)*f1*f2+(-1/16)*f1*f3+(-1/16)*f1*f4+(-1/
16)*f2*f3+(-1/16)*f2*f4+(1/16)*f3*f4+(1/16)*f1*f2*f3+(1/16)*f1*f2*f4+(-1/
16)*f1*f3*f4+(-1/16)*f2*f3*f4+(1/16)*f1*f2*f3*f4,
(1/16)*<identity> of ...+(-1/16)*f1+(1/16)*f2+(1/16)*f3+(1/16)*f4+(-1/
16)*f1*f2+(-1/16)*f1*f3+(-1/16)*f1*f4+(1/16)*f2*f3+(1/16)*f2*f4+(1/
16)*f3*f4+(-1/16)*f1*f2*f3+(-1/16)*f1*f2*f4+(-1/16)*f1*f3*f4+(1/
16)*f2*f3*f4+(-1/16)*f1*f2*f3*f4, (1/16)*<identity> of ...+(1/16)*f1+(-1/
16)*f2+(1/16)*f3+(1/16)*f4+(-1/16)*f1*f2+(1/16)*f1*f3+(1/16)*f1*f4+(-1/
16)*f2*f3+(-1/16)*f2*f4+(1/16)*f3*f4+(-1/16)*f1*f2*f3+(-1/16)*f1*f2*f4+(1/
16)*f1*f3*f4+(-1/16)*f2*f3*f4+(-1/16)*f1*f2*f3*f4,
(1/4)*<identity> of ...+(-1/4)*f3+(1/4)*f4+(-1/4)*f3*f4,
(1/2)*<identity> of ...+(-1/2)*f4 ]
gap> QG := GroupRing( Rationals, SmallGroup(24,12) );;
gap> PrimitiveCentralIdempotentsByESSP( QG );
Wedderga: Warning!!!
The output is a NON-COMPLETE list of prim. central idemp.s of the input!
[ (1/24)*<identity> of ...+(1/24)*f1+(1/24)*f2+(1/24)*f3+(1/24)*f4+(1/
24)*f1*f2+(1/24)*f1*f3+(1/24)*f1*f4+(1/24)*f2^2+(1/24)*f2*f3+(1/
24)*f2*f4+(1/24)*f3*f4+(1/24)*f1*f2^2+(1/24)*f1*f2*f3+(1/24)*f1*f2*f4+(1/
24)*f1*f3*f4+(1/24)*f2^2*f3+(1/24)*f2^2*f4+(1/24)*f2*f3*f4+(1/24)*f1*f2^
2*f3+(1/24)*f1*f2^2*f4+(1/24)*f1*f2*f3*f4+(1/24)*f2^2*f3*f4+(1/24)*f1*f2^
2*f3*f4, (1/24)*<identity> of ...+(-1/24)*f1+(1/24)*f2+(1/24)*f3+(1/
24)*f4+(-1/24)*f1*f2+(-1/24)*f1*f3+(-1/24)*f1*f4+(1/24)*f2^2+(1/
24)*f2*f3+(1/24)*f2*f4+(1/24)*f3*f4+(-1/24)*f1*f2^2+(-1/24)*f1*f2*f3+(-1/
24)*f1*f2*f4+(-1/24)*f1*f3*f4+(1/24)*f2^2*f3+(1/24)*f2^2*f4+(1/
24)*f2*f3*f4+(-1/24)*f1*f2^2*f3+(-1/24)*f1*f2^2*f4+(-1/24)*f1*f2*f3*f4+(1/
24)*f2^2*f3*f4+(-1/24)*f1*f2^2*f3*f4, (1/6)*<identity> of ...+(-1/12)*f2+(
1/6)*f3+(1/6)*f4+(-1/12)*f2^2+(-1/12)*f2*f3+(-1/12)*f2*f4+(1/6)*f3*f4+(-1/
12)*f2^2*f3+(-1/12)*f2^2*f4+(-1/12)*f2*f3*f4+(-1/12)*f2^2*f3*f4 ]



##### 4.3-2 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 (9.15) 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 $$K$$ (9.19).

If the list of primitive central idempotents given by the output is not complete (i.e. if the group $$G$$ is not strongly monomial (9.17)) 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!



##### 4.3-3 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.


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 (4.3-2), 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



#### 4.4 Complete set of orthogonal primitive idempotents from Shoda pairs and cyclotomic classes

##### 4.4-1 PrimitiveIdempotentsNilpotent
 ‣ PrimitiveIdempotentsNilpotent( FG, H, K, C, args ) ( operation )

Returns: A list of orthogonal primitive idempotents.

The input FG should be a semisimple group algebra of a finite nilpotent group $$G$$ whose coefficient field $$F$$ is a finite field. H and K should form a strong Shoda pair $$(H,K)$$ of $$G$$. args is a list containing an epimorphism map epi from $$N_G(K)$$ to $$N_G(K)/K$$ and a generator gq of $$H/K$$. $$C$$ is the $$|F|$$-cyclotomic class modulo $$[H:K]$$ (w.r.t. the generator $$gq$$ of $$H/K$$)

The output is a complete set of orthogonal primitive idempotents of the simple algebra $$FGe_C(G,H,K)$$ (9.22).


gap> G:=DihedralGroup(8);;
gap> F:=GF(3);;
gap> FG:=GroupRing(F,G);;
gap> H:=StrongShodaPairs(G)[5][1];
Group([ f1*f2*f3, f3 ])
gap> K:=StrongShodaPairs(G)[5][2];
Group([ f1*f2 ])
gap> N:=Normalizer(G,K);
Group([ f1*f2*f3, f3 ])
gap> epi:=NaturalHomomorphismByNormalSubgroup(N,K);
[ f1*f2*f3, f3 ] -> [ f1, f1 ]
gap> QHK:=Image(epi,H);
Group([ f1, f1 ])
gap> gq:=MinimalGeneratingSet(QHK)[1];
f1
gap> C:=CyclotomicClasses(Size(F),Index(H,K))[2];
[ 1 ]
gap> PrimitiveIdempotentsNilpotent(FG,H,K,C,[epi,gq]);
[ (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3)^0)*f1*f2+(Z(3))*f1*f2*f3,
(Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3))*f1*f2+(Z(3)^0)*f1*f2*f3 ]



##### 4.4-2 PrimitiveIdempotentsTrivialTwisting
 ‣ PrimitiveIdempotentsTrivialTwisting( FG, H, K, C, args ) ( operation )

Returns: A list of orthogonal primitive idempotents.

The input FG should be a semisimple group algebra of a finite group $$G$$ whose coefficient field $$F$$ is a finite field. H and K should form a strong Shoda pair $$(H,K)$$ of $$G$$. args is a list containing an epimorphism map epi from $$N_G(K)$$ to $$N_G(K)/K$$ and a generator gq of $$H/K$$. $$C$$ is the $$|F|$$-cyclotomic class modulo $$[H:K]$$ (w.r.t. the generator $$gq$$ of $$H/K$$). The input parameters should be such that the simple component $$FGe_C(G,H,K)$$ has a trivial twisting.

The output is a complete set of orthogonal primitive idempotents of the simple algebra $$FGe_C(G,H,K)$$ (9.22).


gap> G:=DihedralGroup(8);;
gap> F:=GF(3);;
gap> FG:=GroupRing(F,G);;
gap> H:=StrongShodaPairs(G)[5][1];
Group([ f1*f2*f3, f3 ])
gap> K:=StrongShodaPairs(G)[5][2];
Group([ f1*f2 ])
gap> N:=Normalizer(G,K);
Group([ f1*f2*f3, f3 ])
gap> epi:=NaturalHomomorphismByNormalSubgroup(N,K);
[ f1*f2*f3, f3 ] -> [ f1, f1 ]
gap> QHK:=Image(epi,H);
Group([ f1, f1 ])
gap> gq:=MinimalGeneratingSet(QHK)[1];
f1
gap> C:=CyclotomicClasses(Size(F),Index(H,K))[2];
[ 1 ]
gap> PrimitiveIdempotentsTrivialTwisting(FG,H,K,C,[epi,gq]);
[ (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3)^0)*f1*f2+(Z(3))*f1*f2*f3,
(Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3))*f1*f2+(Z(3)^0)*f1*f2*f3 ]


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