Multigraded strong Lefschetz property for balanced simplicial complexes

Generalizing the strong Lefschetz property for an N-graded algebra, we introduce the multigraded strong Lefschetz property for an Nᵐ-graded algebra. We show that, for a Nᵐ_+, 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 h₁ h₂ for b c a-b among the flag h-numbers 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 the balanced generalized lower bound inequality by Juhnke-Kubitzke and Murai. We further generalize these results to a-balanced homology manifolds and a-balanced simplicial cycles over a field of characteristic 2.