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])