Stanley’s nonunimodal Gorenstein $h$-vector is optimal
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- by Juan Migliore and Fabrizio Zanello PDF
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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.References
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Additional Information
- Juan Migliore
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- MR Author ID: 124490
- ORCID: 0000-0001-5528-4520
- Email: migliore.1@nd.edu
- Fabrizio Zanello
- Affiliation: Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931
- MR Author ID: 721303
- Email: zanello@mtu.edu
- Received by editor(s): December 4, 2015
- Published electronically: September 29, 2016
- Communicated by: Irena Peeva
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1-9
- MSC (2010): Primary 13D40; Secondary 13H10, 13E10, 05E40
- DOI: https://doi.org/10.1090/proc/13381
- MathSciNet review: 3565355