Weak Lefschetz Property for Boundaries of Simplicial Polytopes

We investigate the weak Lefschetz property for Artinian reductions A = K[∆]/(θ) of the StanleyReisner ring K[∆] of a simplicial complex ∆ by a linear system of parameters θ = (θ1, . . . , θd). It is known that deciding whether or not the ring A is Artinian is matroidal, in the sense that it is controlled only by the matroid M(θ) for the coefficient matrix specifying θ. We consider the weak Lefschetz property for a degree one element ` in K[∆], meaning that multiplication by ` from Ai → Ai+1 has full rank for all i, and ask when it is matroidal in the sense of of being controlled only by the lifted matroid M̂(θ, `) for the coefficient matrix of θ and `. It is known from work of Stanley [3], McMullen and others that when K = R and ∆ is the boundary of a simplicial polytope, there exists a choice θ and degree one elements ` having the weak Lefschetz property. In this context, we find some examples where the weak Lefschetz property is not matroidal, and others where it is matroidal.