Stanley’s nonunimodal Gorenstein ℎ-vector is optimal

Migliore, Juan; Zanello, Fabrizio

doi:10.1090/proc/13381

Stanley’s nonunimodal Gorenstein $h$-vector is optimal
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Proc. Amer. Math. Soc. 145 (2017), 1-9 Request permission

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 longstanding open question in this area. All of our results are characteristic free.

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