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Multigraded strong Lefschetz property for balanced simplicial complexes
Oba, Ryoshun1
1 University of Tokyo Department of Mathematical Informatics Graduate School of Information Science and Technology 7-3-1 Hongo Bunkyo-ku 113-8656 Tokyo Japan
Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 775-794.

Generalizing the strong Lefschetz property for $\mathbb{N}$-graded algebras, we introduce the multigraded strong Lefschetz property. We show that, for $\mathbf{a} \in \mathbb{N}^m_+$, the generic $\mathbb{N}^m$-graded Artinian reduction of the Stanley-Reisner ring of an $\mathbf{a}$-balanced homology sphere over a field of characteristic $2$ satisfies the multigraded strong Lefschetz property. As a corollary, we prove that the flag $h$-numbers of an $\mathbf{a}$-balanced simplicial sphere satisfy $h_{\mathbf{b}} \le h_{\mathbf{c}}$ for $\mathbf{b} \le \mathbf{c} \le \mathbf{a}-\mathbf{b}$. This result can be viewed as a common generalization of the unimodality of the $h$-vector of a simplicial sphere by Adiprasito and the balanced generalized lower bound inequality by Juhnke-Kubitzke and Murai. We further generalize these results to $\mathbf{a}$-balanced homology manifolds and $\mathbf{a}$-balanced simplicial cycles over fields of characteristic $2$.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.419
Classification: 13F55
Keywords: Lefschetz property, Stanley-Reisner ring, balancedness, multigraded algebra, unimodality

Oba, Ryoshun 1

1 University of Tokyo Department of Mathematical Informatics Graduate School of Information Science and Technology 7-3-1 Hongo Bunkyo-ku 113-8656 Tokyo Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Oba, Ryoshun. Multigraded strong Lefschetz property for balanced simplicial complexes. Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 775-794. doi : 10.5802/alco.419. https://alco.centre-mersenne.org/articles/10.5802/alco.419/

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