Multigraded strong Lefschetz property for balanced simplicial complexes

TL;DR

This paper introduces a multigraded version of the strong Lefschetz property for balanced simplicial complexes, proving it for certain algebraic reductions and deriving inequalities for flag h-numbers, thus unifying previous results on unimodality and lower bounds.

Contribution

It defines the multigraded strong Lefschetz property for $ $-graded algebras and proves it for balanced homology spheres over characteristic 2, generalizing key combinatorial inequalities.

Findings

Proves the multigraded strong Lefschetz property for certain algebraic reductions.

Derives inequalities among flag h-numbers of balanced simplicial spheres.

Unifies unimodality and lower bound inequalities in a common framework.

Abstract

Generalizing the strong Lefschetz property for an $\mathbb{N}$-graded algebra, we introduce the multigraded strong Lefschetz property for an $\mathbb{N}^m$-graded algebra. 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. A corollary is the inequality $h_{\mathbf{b}} \leq h_{\mathbf{c}}$ for $\mathbf{b} \leq \mathbf{c} \leq \mathbf{a}-\mathbf{b}$ among the flag $h$-numbers of an $\mathbf{a}$-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 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 a field of characteristic $2$.