Type-B generalized triangulations and determinantal ideals

For n≥3n≥3, let ΩnΩn be the set of line segments between the vertices of a convex nn-gon. For j≥2j≥2, a jj-crossing is a set of jj line segments pairwise intersecting in the relative interior of the nn-gon. For k≥1k≥1, let Δn,kΔn,k be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of ΩnΩn not containing any (k+1)(k+1)-crossing.The complex Δn,kΔn,k has been the central object of many papers. Here we continue this work by considering the complex of type-B generalized triangulations. For this we identify line segments in Ω2nΩ2n which can be transformed into each other by a 180∘-rotation of the 2n2n-gon. Let FnFn be the set Ω2nΩ2n after identification, then the complex Dn,kDn,k of type-B generalized triangulations is the simplicial complex of subsets of FnFn not containing any (k+1)(k+1)-crossing in the above sense. For k=1k=1, we have that Dn,1Dn,1 is the simplicial complex of type-B triangulations of the 2n2n-gon as defined in [R. Simion, A type-B associahedron, Adv. Appl. Math. 30 (2003) 2–25] and decomposes into a join of an (n−1)(n−1)-simplex and the boundary of the nn-dimensional cyclohedron. We demonstrate that Dn,kDn,k is a pure, k(n−k)−1+knk(n−k)−1+kn dimensional complex that decomposes into a kn−1kn−1-simplex and a k(n−k)−1k(n−k)−1 dimensional homology-sphere. For k=n−2k=n−2 we show that this homology-sphere is in fact the boundary of a cyclic polytope. We provide a lower and an upper bound for the number of maximal faces of Dn,kDn,k.On the algebraical side we give a term order on the monomials in the variables Xij,1≤i,j≤nXij,1≤i,j≤n, such that the corresponding initial ideal of the determinantal ideal generated by the (k+1)(k+1) times (k+1)(k+1) minors of the generic n×nn×n matrix contains the Stanley–Reisner ideal of Dn,kDn,k. We show that the minors form a Gröbner-Basis whenever k∈{1,n−2,n−1}k∈{1,n−2,n−1} thereby proving the equality of both ideals and the unimodality of the hh-vector of the determinantal ideal in these cases. We conjecture this result to be true for all values of k