On Two Classes of Closely Related Monomial Ideals

TL;DR

This paper establishes an equivalence between the Hilbert depth formulas of squarefree Veronese ideals and powers of the irrelevant maximal ideal, revealing a deep connection and conjecturing similar equivalence for Stanley depths.

Contribution

It proves the equivalence of Hilbert depth formulas for two classes of monomial ideals and suggests a potential link in their Stanley depths.

Findings

Hilbert depth formulas are equivalent for the two classes of ideals

Revealed a strong connection between seemingly unrelated monomial ideals

Conjectured Stanley depths are also equivalent

Abstract

In [7] we obtained a formula for the Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring by relating it to the Hilbert depth of powers of the irrelevant maximal ideal. In this paper, we prove that these two Hilbert depth formulas are equivalent to each other. Our result reveals that there is a strong connection between these two classes of seemingly unrelated monomial ideals. We conjecture that their Stanley depths are equivalent as well.