Unimodal Gorenstein h-vectors without the Stanley-Iarrobino property

TL;DR

This paper investigates unimodal Gorenstein h-vectors that do not satisfy the Stanley-Iarrobino property, establishing their existence in certain socle degrees and codimensions, and conjecturing their independence from field characteristic.

Contribution

It demonstrates the existence of such h-vectors in socle degree at least 6 and in codimension five or higher, advancing understanding of Gorenstein algebra properties.

Findings

Existence of unimodal Gorenstein h-vectors without Stanley-Iarrobino property for socle degree ≥6

Such h-vectors exist in every codimension ≥5

Open problem remains for codimension 4 cases

Abstract

The study of the $h$-vectors of graded Gorenstein algebras is an important topic in combinatorial commutative algebra, which despite the large amount of literature produced during the last several years, still presents many interesting open questions. In this note, we commence a study of those unimodal Gorenstein $h$-vectors that do \emph{not} satisfy the Stanley-Iarrobino property. Our main results, which are characteristic free, show that such $h$-vectors exist: 1) In socle degree $e$ if and only if $e\ge 6$; and 2) In every codimension five or greater. The main case that remains open is that of codimension four, where no Gorenstein $h$-vector is known without the Stanley-Iarrobino property. We conclude by proposing the following very general conjecture: The existence of any arbitrary level $h$-vector is \emph{independent} of the characteristic of the base field.