Multigraded strong Lefschetz property for balanced simplicial complexes

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 $2$.