Stanley's nonunimodal Gorenstein h-vector is optimal

TL;DR

This paper classifies all possible Gorenstein h-vectors in specific socle degrees and codimensions, establishing the minimal variables needed for nonunimodal examples and solving a long-standing open problem.

Contribution

It provides a complete classification of Gorenstein h-vectors in socle degrees 4 and 5 for certain codimensions, identifying the minimal variables for nonunimodality.

Findings

The smallest nonunimodal Gorenstein h-vector is (1,13,12,13,1).

Nonunimodal Gorenstein h-vectors exist starting from 13 variables in socle degree 4.

The results are characteristic free and settle a long-standing open question.

Abstract

We classify all possible $h$-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension $\leq 17$, and in socle degree 5 and codimension $\leq 25$. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein $h$-vector is 13 for socle degree 4, and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein $h$-vector is $(1,13,12,13,1)$, which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a long-standing open question in this area. All of our results are characteristic free.