An intriguing ring structure on the set of d-forms

The purpose of this note is to introduce a multiplication on the set of homogeneous polynomials of fixed degree d, in a way to provide a duality theory between monomial ideals of K[x_1,\ldots,x_d] generated in degrees \leq n and block stable ideals (a class of ideals containing the Borel fixed ones) of K[x_1,\ldots,x_n] generated in degree d. As a byproduct we give a new proof of the characterization of Betti tables of ideals with linear resolution given by Murai.