An intriguing ring structure on the set of d-forms

TL;DR

This paper introduces a new multiplication on homogeneous polynomials to establish a duality between certain monomial ideals and block stable ideals, providing a novel perspective on Betti tables of ideals with linear resolutions.

Contribution

It defines a multiplication on d-forms that creates a duality theory linking monomial ideals and block stable ideals, offering new insights into their algebraic properties.

Findings

Establishes a duality between monomial and block stable ideals.

Provides a new proof of Murai's characterization of Betti tables.

Introduces a ring structure on the set of d-forms.

Abstract

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.