The following examples demonstrate the internalization of the symmetric algera and its modules in the positively graded closure of various tensor categories.
gap> F2 := HomalgRingOfIntegers( 2 );; gap> F2vec := MatrixCategory( F2 );; gap> v := VectorSpaceObject( 3, F2 );; gap> ZF2vec := PositivelyZGradedClosureCategory( F2vec ); PositivelyZGradedClosureCategory( Category of matrices over GF(2) ) gap> V := ObjectInZGradedClosureCategoryWithBounds( ZF2vec, v, 1 );; gap> SVMod := CategoryOfLeftSModules( v ); Abelian category of left modules over the internal symmetric algebra of A vector space object over GF(2) of dimension 3 with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts gap> ModSV := CategoryOfRightSModules( v );; gap> SV := UnderlyingActingObject( SVMod ); <An object in PositivelyZGradedClosureCategory( Category of matrices over GF(2) )> gap> S := SymmetricAlgebraAsLeftModule( v ); <An object in Abelian category of left modules over the internal symmetric algebra of A vector space object over GF(2) of dimension 3 with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts> gap> u := VectorSpaceObject( 3, F2 );; gap> U := ObjectInZGradedClosureCategoryWithBounds( ZF2vec, u, 3 );; gap> F := FreeInternalModule( U, SVMod ); <An object in Abelian category of left modules over the internal symmetric algebra of A vector space object over GF(2) of dimension 3 with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts> gap> H := FreeInternalModule( U, ModSV );; gap> e1 := MorphismInZGradedClosureCategoryWithBounds( > ObjectInZGradedClosureCategoryWithBounds( ZF2vec, SV[0], 3 ), > VectorSpaceMorphism( > SV[0], > CertainRows( HomalgIdentityMatrix( 10, F2 ), [ 1 ] ), > SV[3] > ), > 3, > SV );; gap> e1 := InternalElement( e1 ); degree: 3 1 . . . . . . . . . A morphism in Category of matrices over GF(2)
gap> Q := HomalgFieldOfRationals( );; gap> Qvec := MatrixCategory( Q );; gap> v := VectorSpaceObject( 3, Q );; gap> ZQvec := PositivelyZGradedClosureCategory( Qvec ); PositivelyZGradedClosureCategory( Category of matrices over Q ) gap> V := ObjectInZGradedClosureCategoryWithBounds( ZQvec, v, 1 );; gap> SVMod := CategoryOfLeftSModules( v ); Abelian category of left modules over the internal symmetric algebra of A vector space object over Q of dimension 3 with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts gap> ModSV := CategoryOfRightSModules( v );; gap> SV := UnderlyingActingObject( SVMod ); <An object in PositivelyZGradedClosureCategory( Category of matrices over Q )> gap> S := SymmetricAlgebraAsLeftModule( v ); <An object in Abelian category of left modules over the internal symmetric algebra of A vector space object over Q of dimension 3 with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts> gap> u := VectorSpaceObject( 3, Q );; gap> U := ObjectInZGradedClosureCategoryWithBounds( ZQvec, u, 3 );; gap> F := FreeInternalModule( U, SVMod ); <An object in Abelian category of left modules over the internal symmetric algebra of A vector space object over Q of dimension 3 with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts> gap> H := FreeInternalModule( U, ModSV );; gap> e1 := MorphismInZGradedClosureCategoryWithBounds( > ObjectInZGradedClosureCategoryWithBounds( ZQvec, SV[0], 3 ), > VectorSpaceMorphism( > SV[0], > CertainRows( HomalgIdentityMatrix( 10, Q ), [ 1 ] ), > SV[3] > ), > 3, > SV );; gap> e1 := InternalElement( e1 ); degree: 3 [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ] A morphism in Category of matrices over Q
gap> G := SymmetricGroup( 4 );; gap> RepG := RepresentationCategory( G );; gap> irr := Irr( G );; gap> one := RepresentationCategoryObject( irr[5], RepG, "1" );; gap> sigma := RepresentationCategoryObject( irr[1], RepG, "sigma" );; gap> rho := RepresentationCategoryObject( irr[3], RepG, "rho" );; gap> nu := RepresentationCategoryObject( irr[4], RepG, "nu" );; gap> chi := RepresentationCategoryObject( irr[2], RepG, "chi" );; gap> v := chi;; gap> ZRepG := PositivelyZGradedClosureCategory( RepG ); PositivelyZGradedClosureCategory( The representation category of SymmetricGroup( [ 1 .. 4 ] ) ) gap> V := ObjectInZGradedClosureCategoryWithBounds( ZRepG, v, 1 );; gap> SVMod := CategoryOfLeftSModules( v ); Abelian category of left modules over the internal symmetric algebra of 1*(chi) with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts gap> ModSV := CategoryOfRightSModules( v );; gap> SV := UnderlyingActingObject( SVMod ); <An object in PositivelyZGradedClosureCategory( The representation category of SymmetricGroup( [ 1 .. 4 ] ) )> gap> S := SymmetricAlgebraAsLeftModule( v ); <An object in Abelian category of left modules over the internal symmetric algebra of 1*(chi) with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts> gap> u := chi;; gap> U := ObjectInZGradedClosureCategoryWithBounds( ZRepG, u, 3 );; gap> F := FreeInternalModule( U, SVMod ); <An object in Abelian category of left modules over the internal symmetric algebra of 1*(chi) with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts> gap> H := FreeInternalModule( U, ModSV );; gap> c1 := UniversalMorphismFromFreeModule( F, Support( F[6] )[4], 6, 1 );; gap> c2 := UniversalMorphismFromFreeModule( F, Support( F[6] )[4], 6, 2 );; gap> c := 2 * c1 - 3 * c2;; gap> Display( c[6] ); Component: (nu) 2,-3,0,0 A morphism in Category of matrices over Q ------------------------ gap> a := InternalElement( SV, Support( SV[1] )[1], 1, 1 );; gap> b := InternalElement( SV, Support( SV[2] )[1], 2, 1 );; gap> c := InternalElement( SV, Support( SV[2] )[2], 2, 1 );; gap> d := InternalElement( SV, Support( SV[3] )[3], 3, 1 );; gap> cc := UniversalMorphismFromFreeModule( c );; gap> dd := UniversalMorphismFromFreeModule( d );; gap> pp := ProjectionInFactorOfFiberProduct( [ cc, dd ], 1 );; gap> qq := ProjectionInFactorOfFiberProduct( [ cc, dd ], 2 );; gap> ff := PreCompose( pp, cc ) - PreCompose( qq, dd );; gap> Set( List( Sublist( ff, [ 0 .. 5 ] ), IsZero ) ); [ true ] gap> ss := UniversalMorphismIntoDirectSum( [ pp, -qq ] );; gap> tt := UniversalMorphismFromDirectSum( [ cc, dd ] );; gap> homology := HomologyObject( ss, tt );; gap> Set( List( Sublist( homology, [ 0 .. 5 ] ), IsZero ) ); [ true ] gap> e1 := InternalElement( SV, sigma, 3, 1 );; gap> e2 := InternalElement( SV, chi, 3, 1 );; gap> e3 := InternalElement( SV, chi, 3, 2 );; gap> e4 := InternalElement( SV, nu, 3, 1 );; gap> e := e1 + 2 * e2 - 1/3 * e3 + e4; degree: 3 Component: (sigma) 1 A morphism in Category of matrices over Q ------------------------ Component: (chi) 2,-1/3 A morphism in Category of matrices over Q ------------------------ Component: (nu) 1 A morphism in Category of matrices over Q ------------------------ gap> e2 * e3 = BraidedMultiplication( e3, e2 ); true gap> m1 := InternalElement( F, nu, 6, 1 );; gap> m2 := InternalElement( F, nu, 6, 2 );; gap> m := 2 * m1 - 1/3 * m2; degree: 6 Component: (nu) 2,-1/3,0,0 A morphism in Category of matrices over Q ------------------------
gap> G := SymmetricGroup( 4 );; gap> RepG := RepresentationCategoryZGraded( G );; gap> irr := Irr( G );; gap> v := RepresentationCategoryZGradedObject( 1, irr[2], RepG );; gap> ZRepG := PositivelyZGradedClosureCategory( RepG ); PositivelyZGradedClosureCategory( The skeletal Z-graded representation category of SymmetricGroup( [ 1 .. 4 ] ) ) gap> V := ObjectInZGradedClosureCategoryWithBounds( ZRepG, v );; gap> SVMod := CategoryOfLeftSModules( v ); Abelian category of left modules over the internal symmetric algebra of 1*(x_[1, 2]) with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts gap> ModSV := CategoryOfRightSModules( v );; gap> SV := UnderlyingActingObject( SVMod ); <An object in PositivelyZGradedClosureCategory( The skeletal Z-graded representation category of SymmetricGroup( [ 1 .. 4 ] ) )> gap> S := SymmetricAlgebraAsLeftModule( v ); <An object in Abelian category of left modules over the internal symmetric algebra of 1*(x_[1, 2]) with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts> gap> u := RepresentationCategoryZGradedObject( 3, irr[2], RepG );; gap> U := ObjectInZGradedClosureCategoryWithBounds( ZRepG, u );; gap> F := FreeInternalModule( U, SVMod ); <An object in Abelian category of left modules over the internal symmetric algebra of 1*(x_[1, 2]) with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts> gap> H := FreeInternalModule( U, ModSV );; gap> nu6 := Support( F[6] )[4]; <x_[6, 4]> gap> c1 := UniversalMorphismFromFreeModule( F, nu6, 1 );; gap> c2 := UniversalMorphismFromFreeModule( F, nu6, 2 );; gap> c := 2 * c1 - 3 * c2;; gap> Display( c[6] ); Component: (x_[6, 4]) 2,-3,0,0 A morphism in Category of matrices over Q ------------------------ gap> a := InternalElement( SV, Support( SV[1] )[1], 1 );; gap> b := InternalElement( SV, Support( SV[2] )[1], 1 );; gap> c := InternalElement( SV, Support( SV[2] )[2], 1 );; gap> d := InternalElement( SV, Support( SV[3] )[3], 1 );; gap> cc := UniversalMorphismFromFreeModule( c );; gap> dd := UniversalMorphismFromFreeModule( d );; gap> pp := ProjectionInFactorOfFiberProduct( [ cc, dd ], 1 );; gap> qq := ProjectionInFactorOfFiberProduct( [ cc, dd ], 2 );; gap> ff := PreCompose( pp, cc ) - PreCompose( qq, dd );; gap> Set( List( Sublist( ff, [ 0 .. 5 ] ), IsZero ) ); [ true ] gap> ss := UniversalMorphismIntoDirectSum( [ pp, -qq ] );; gap> tt := UniversalMorphismFromDirectSum( [ cc, dd ] );; gap> homology := HomologyObject( ss, tt );; gap> Set( List( Sublist( homology, [ 0 .. 5 ] ), IsZero ) ); [ true ] gap> sigma3 := Support( SV[3] )[1]; <x_[3, 1]> gap> chi3 := Support( SV[3] )[2]; <x_[3, 2]> gap> nu3 := Support( SV[3] )[3]; <x_[3, 4]> gap> e1 := InternalElement( SV, sigma3, 1 );; gap> e2 := InternalElement( SV, chi3, 1 );; gap> e3 := InternalElement( SV, chi3, 2 );; gap> e4 := InternalElement( SV, nu3, 1 );; gap> e := e1 + 2 * e2 - 1/3 * e3 + e4; degree: 3 Component: (x_[3, 1]) 1 A morphism in Category of matrices over Q ------------------------ Component: (x_[3, 2]) 2,-1/3 A morphism in Category of matrices over Q ------------------------ Component: (x_[3, 4]) 1 A morphism in Category of matrices over Q ------------------------ gap> e2 * e3 = BraidedMultiplication( e3, e2 ); true gap> m1 := InternalElement( F, nu6, 1 );; gap> m2 := InternalElement( F, nu6, 2 );; gap> m := 2 * m1 - 1/3 * m2; degree: 6 Component: (x_[6, 4]) 2,-1/3,0,0 A morphism in Category of matrices over Q ------------------------
This is an example for explicit computations in the symmetric algebra \(\mathrm{S}W\) internal to the positively graded closure \(\mathrm{srep}_k(G)_+\) of the tensor category \(\mathrm{srep}_k(G)\), where \(G := S_4\) is the symmetric group on four points and \(k = \mathbb{Q}\), its minimal splitting field.
gap> LoadPackage( "InternalModules" : OnlyNeeded ); true gap> srepG := RepresentationCategory( SymmetricGroup( 4 ) ); The representation category of SymmetricGroup( [ 1 .. 4 ] ) gap> Display( srepG ); A CAP category with name The representation category of SymmetricGroup( [ 1 .. 4 ] ): 44 primitive operations were used to derive 357 operations for this category which algorithmically * IsEquippedWithHomomorphismStructure * IsLinearCategoryOverCommutativeRing * IsAbelianCategory * IsRigidSymmetricClosedMonoidalCategory and furthermore mathematically * IsSkeletalCategory gap> G := UnderlyingGroupForRepresentationCategory( srepG ); Sym( [ 1 .. 4 ] ) gap> irr := Irr( G );; gap> one := RepresentationCategoryObject( irr[5], srepG, "𝟙" ); 1*(𝟙) gap> sigma := RepresentationCategoryObject( irr[1], srepG, "σ" ); 1*(σ) gap> rho := RepresentationCategoryObject( irr[3], srepG, "ρ" ); 1*(ρ) gap> nu := RepresentationCategoryObject( irr[4], srepG, "ν" ); 1*(ν) gap> chi := RepresentationCategoryObject( irr[2], srepG, "χ" ); 1*(χ) gap> TensorProduct( rho, sigma, rho ); 1*(σ) + 1*(ρ) + 1*(𝟙) gap> TensorProduct( rho, rho ); 1*(σ) + 1*(ρ) + 1*(𝟙) gap> Display( AssociatorRightToLeft( rho, sigma, rho ) ); Component: (σ) -1 A morphism in Category of matrices over Q ------------------------ Component: (ρ) 1 A morphism in Category of matrices over Q ------------------------ Component: (𝟙) -1 A morphism in Category of matrices over Q ------------------------ gap> Display( Braiding( rho, rho ) ); Component: (σ) -1 A morphism in Category of matrices over Q ------------------------ Component: (ρ) 1 A morphism in Category of matrices over Q ------------------------ Component: (𝟙) 1 A morphism in Category of matrices over Q ------------------------ gap> W := chi; 1*(χ) gap> LoadPackage( "GradedCategories" ); true gap> ZsrepG := PositivelyZGradedClosureCategory( srepG ); PositivelyZGradedClosureCategory( The representation category of SymmetricGroup( [ 1 .. 4 ] ) ) gap> LoadPackage( "InternalModules" ); true gap> SWMod := CategoryOfLeftSModules( W ); Abelian category of left modules over the internal symmetric algebra of 1*(χ) with undecidable (mathematical) equality of morphisms and uncomputable lifts and colifts gap> SW := UnderlyingActingObject( SWMod ); <An object in PositivelyZGradedClosureCategory( The representation category of SymmetricGroup( [ 1 .. 4 ] ) )> gap> SW[0]; 1*(𝟙) gap> SW[1]; 1*(χ) gap> SW[2]; 1*(ρ) + 1*(ν) + 1*(𝟙) gap> SW[3]; 1*(σ) + 2*(χ) + 1*(ν) gap> SW[4]; 1*(χ) + 2*(ρ) + 2*(ν) + 2*(𝟙) gap> chi1 := InternalElement( SW, chi, 1, 1 ); degree: 1 Component: (χ) 1 A morphism in Category of matrices over Q ------------------------ gap> rho2 := InternalElement( SW, rho, 2, 1 ); degree: 2 Component: (ρ) 1 A morphism in Category of matrices over Q ------------------------ gap> nu2 := InternalElement( SW, nu, 2, 1 ); degree: 2 Component: (ν) 1 A morphism in Category of matrices over Q ------------------------ gap> nu3 := InternalElement( SW, nu, 3, 1 ); degree: 3 Component: (ν) 1 A morphism in Category of matrices over Q ------------------------ gap> chi1 * chi1; degree: 2 Component: (ρ) 1 A morphism in Category of matrices over Q ------------------------ Component: (ν) 1 A morphism in Category of matrices over Q ------------------------ Component: (𝟙) 1 A morphism in Category of matrices over Q ------------------------ gap> chi1 * (chi1 * chi1); degree: 3 Component: (σ) 1 A morphism in Category of matrices over Q ------------------------ Component: (χ) 5,3 A morphism in Category of matrices over Q ------------------------ Component: (ν) 3 A morphism in Category of matrices over Q ------------------------ gap> chi1 * (chi1 * chi1) = (chi1 * chi1) * chi1; true gap> chi1 * rho2; degree: 3 Component: (χ) 0,2 A morphism in Category of matrices over Q ------------------------ Component: (ν) 1 A morphism in Category of matrices over Q ------------------------ gap> chi1 * rho2 = rho2 * chi1; true gap> chi1 * nu2; degree: 3 Component: (σ) 1 A morphism in Category of matrices over Q ------------------------ Component: (χ) 8,0 A morphism in Category of matrices over Q ------------------------ Component: (ν) 2 A morphism in Category of matrices over Q ------------------------ gap> chi1 * nu2 = nu2 * chi1; true gap> chi1 * (chi1 * (chi1 * chi1)); degree: 4 Component: (χ) -2 A morphism in Category of matrices over Q ------------------------ Component: (ρ) 3,-31 A morphism in Category of matrices over Q ------------------------ Component: (ν) 5,6 A morphism in Category of matrices over Q ------------------------ Component: (𝟙) 5,3 A morphism in Category of matrices over Q ------------------------ gap> (chi1 * chi1) * (chi1 * chi1); degree: 4 Component: (χ) 8 A morphism in Category of matrices over Q ------------------------ Component: (ρ) 9/4,43/4 A morphism in Category of matrices over Q ------------------------ Component: (ν) -16,-6 A morphism in Category of matrices over Q ------------------------ Component: (𝟙) -7,9/8 A morphism in Category of matrices over Q ------------------------ gap> ((chi1 * chi1) * chi1) * chi1; degree: 4 Component: (χ) 26 A morphism in Category of matrices over Q ------------------------ Component: (ρ) 3,-31 A morphism in Category of matrices over Q ------------------------ Component: (ν) 5,6 A morphism in Category of matrices over Q ------------------------ Component: (𝟙) 5,3 A morphism in Category of matrices over Q ------------------------ gap> (chi1 * chi1) * rho2; degree: 4 Component: (χ) 4 A morphism in Category of matrices over Q ------------------------ Component: (ρ) 5/4,-9/4 A morphism in Category of matrices over Q ------------------------ Component: (ν) -4,-4 A morphism in Category of matrices over Q ------------------------ Component: (𝟙) 0,1/8 A morphism in Category of matrices over Q ------------------------ gap> chi1 * (chi1 * rho2); degree: 4 Component: (χ) -2 A morphism in Category of matrices over Q ------------------------ Component: (ρ) 2,-12 A morphism in Category of matrices over Q ------------------------ Component: (ν) -1,-1 A morphism in Category of matrices over Q ------------------------ Component: (𝟙) 0,2 A morphism in Category of matrices over Q ------------------------ gap> (chi1 * chi1) * nu2; degree: 4 Component: (χ) 4 A morphism in Category of matrices over Q ------------------------ Component: (ρ) 0,16 A morphism in Category of matrices over Q ------------------------ Component: (ν) -8,-1 A morphism in Category of matrices over Q ------------------------ Component: (𝟙) -4,0 A morphism in Category of matrices over Q ------------------------ gap> chi1 * nu3; degree: 4 Component: (χ) 4 A morphism in Category of matrices over Q ------------------------ Component: (ρ) 0,-12 A morphism in Category of matrices over Q ------------------------ Component: (ν) 1,1 A morphism in Category of matrices over Q ------------------------ gap> chi1 * nu3 = nu3 * chi1; true gap> chi1 * (chi1 * (chi1 * chi1)) <> ((chi1 * chi1) * chi1) * chi1; true
The following examples demonstrate the internalization of the exterior algera and its modules in the finitely graded closure of various tensor categories.
gap> Q := HomalgFieldOfRationals( );; gap> Qvec := MatrixCategory( Q );; gap> v := VectorSpaceObject( 6, Q );; gap> ZQvec := FinitelyZGradedClosureCategory( Qvec ); FinitelyZGradedClosureCategory( Category of matrices over Q ) gap> V := ObjectInZGradedClosureCategoryWithBounds( ZQvec, v, 1 );; gap> ExtVMod := CategoryOfLeftEModules( v ); Abelian category of left modules over the internal exterior algebra of A vector space object over Q of dimension 6 gap> ModExtV := CategoryOfRightEModules( v );; gap> ExtV := UnderlyingActingObject( ExtVMod ); <An object in FinitelyZGradedClosureCategory( Category of matrices over Q )> gap> ext := ExteriorAlgebraAsLeftModule( v ); <An object in Abelian category of left modules over the internal exterior algebra of A vector space object over Q of dimension 6> gap> e1 := MorphismInZGradedClosureCategoryWithBounds( > ObjectInZGradedClosureCategoryWithBounds( ZQvec, ExtV[0], 3 ), > VectorSpaceMorphism( > ExtV[0], > CertainRows( HomalgIdentityMatrix( 20, Q ), [ 6 ] ), > ExtV[3] > ), > 3, > ExtV );; gap> e1 := InternalElement( e1 ); degree: 3 [ [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ] A morphism in Category of matrices over Q gap> u := VectorSpaceObject( 3, Q );; gap> U := ObjectInZGradedClosureCategoryWithBounds( ZQvec, u, 3 );; gap> F := FreeInternalModule( U, ExtVMod ); <An object in Abelian category of left modules over the internal exterior algebra of A vector space object over Q of dimension 6> gap> H := FreeInternalModule( U, ModExtV );;
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