Séminaire Lotharingien de Combinatoire, 93B.56 (2025), 12 pp.
Ryoshun Oba
Multigraded Strong Lefschetz Property for Balanced Simplicial Complexes
Abstract.
Generalizing the strong Lefschetz property for an N-graded algebra, we introduce multigraded strong Lefschetz property for an Nm-graded algebra.
We show that, for a ∈ N+m, the generic Nm-graded Artinian reduction of the Stanley-Reisner ring of an a-balanced homology sphere over a field of characteristic 2 satisfies the multigraded strong Lefschetz property.
A corollary is the inequality hb ≤ hc for b ≤ c ≤ a-b for the flag h-vector of an a-balanced simplicial sphere. This can be seen as a common generalization of the unimodality of the h-vector of a simplicial sphere by Adiprasito and balanced generalized lower bound inequality by Juhnke-Kubitzke and Murai.
Another combinatorial consequence is that a k-dimensional completely balanced simplicial complex which is a subcomplex of a simplicial 2k-sphere satisfies fk ≤ 2fk-1.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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