GAP (Wedderga) - Chapter 4: Idempotents

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4 Idempotents
4.1 Computing idempotents from character table
4.1-1 PrimitiveCentralIdempotentsByCharacterTable
4.2 Testing lists of idempotents for completeness
4.2-1 IsCompleteSetOfOrthogonalIdempotents
4.3 Idempotents from Shoda pairs
4.3-1 PrimitiveCentralIdempotentsByStrongSP
4.3-2 PrimitiveCentralIdempotentsBySP
4.4 Complete set of orthogonal primitive idempotents from Shoda pairs and cyclotomic classes
4.4-1 PrimitiveIdempotentsNilpotent
4.4-2 PrimitiveIdempotentsTrivialTwisting

### 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:

⋅ 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

#### 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 (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 (9.17).

If the list of primitive central idempotents given by the output is not complete (i.e. if the group G is not strongly monomial (9.16)) 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!

##### 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 χ(e)ne 0 for some irreducible monomial character χ 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` (4.3-1), 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

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

##### 4.4-1 PrimitiveIdempotentsNilpotent

`‣ PrimitiveIdempotentsNilpotent`( FG, H, K, C, arg ) | ( 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. arg 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) (9.20).

    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([ <identity> of ..., 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, arg ) | ( 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. arg 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) (9.20).

    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([ <identity> of ..., 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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