gap> LoadPackage( "InternalExteriorAlgebraForCAP" ); true gap> LoadPackage( "HomologicalAlgebraForCAP" ); # needed for computing spectral sequences true gap> RepG := RepresentationCategoryZGraded( SymmetricGroup( 4 ) ); The skeletal Z-graded representation category of SymmetricGroup( [ 1 .. 4 ] ) gap> G := UnderlyingGroupForRepresentationCategory( RepG ); Sym( [ 1 .. 4 ] ) gap> irr := Irr( G ); [ Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, -1, 1, 1, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 3, -1, -1, 0, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 2, 0, 2, -1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 3, 1, -1, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1, 1 ] ) ] gap> v := RepresentationCategoryZGradedObject( -1, irr[2], RepG ); 1*(x_[-1, 2]) gap> cat := EModuleActionCategory( v ); Module category of the internal exterior algebra modeled via right actions of 1*(x_[-1, 2]) gap> h := RepresentationCategoryZGradedObject( 3, irr[2], RepG ); 1*(x_[3, 2]) gap> F := FreeEModule( h, cat ); <An object in Module category of the internal exterior algebra modeled via right actions of 1*(x_[-1, 2])> gap> chi := Support( ActionDomain( F ) )[8]; <x_[2, 4]> gap> c := ComponentInclusionMorphism( ActionDomain( F ), chi ); <A morphism in The skeletal Z-graded representation category of SymmetricGroup( [ 1 .. 4 ] )> gap> u := UniversalMorphismFromFreeModule( F, c ); <A morphism in Module category of the internal exterior algebra modeled via right actions of 1*(x_[-1, 2])> gap> t := FilteredTateResolution( u ); <An object in Cocomplex category of Descending filtered object category of The skeletal Z-graded representation category of SymmetricGroup( [ 1 .. 4 ] )> gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( t, 2, 0, 1 ); <A morphism in Generalized morphism category of The skeletal Z-graded representation category of SymmetricGroup( [ 1 .. 4 ] )> gap> Display( UnderlyingHonestObject( Source( s ) ) ); 1*(x_[2, 3]) + 1*(x_[2, 5])
generated by GAPDoc2HTML