Inspired by a series of conjectures related to higher coinvariant algebras,
we present two families of relations involving harmonic differential forms of
the symmetric group. Our relations, together with a novel bijection, are
sufficient to give a filtration of the 1-forms suggested by work of
Haglund--Rhoades--Shimozono with composition factors given by Tanisaki
quotients. These are "almost all" of the necessary relations in a certain
asymptotic sense we make precise.Comment: 30 page