Type-B generalized triangulations and determinantal ideals

AbstractFor n≥3, let Ωn be the set of line segments between the vertices of a convex n-gon. For j≥2, a j-crossing is a set of j line segments pairwise intersecting in the relative interior of the n-gon. For k≥1, let Δn,k be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of Ωn not containing any (k+1)-crossing.The complex Δn,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 Ω2n which can be transformed into each other by a 180∘-rotation of the 2n-gon. Let Fn be the set Ω2n after identification, then the complex Dn,k of type-B generalized triangulations is the simplicial complex of subsets of Fn not containing any (k+1)-crossing in the above sense. For k=1, we have that Dn,1 is the simplicial complex of type-B triangulations of the 2n-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)-simplex and the boundary of the n-dimensional cyclohedron. We demonstrate that Dn,k is a pure, k(n−k)−1+kn dimensional complex that decomposes into a kn−1-simplex and a k(n−k)−1 dimensional homology-sphere. For k=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,k.On the algebraical side we give a term order on the monomials in the variables Xij,1≤i,j≤n, such that the corresponding initial ideal of the determinantal ideal generated by the (k+1) times (k+1) minors of the generic n×n matrix contains the Stanley–Reisner ideal of Dn,k. We show that the minors form a Gröbner-Basis whenever k∈{1,n−2,n−1} thereby proving the equality of both ideals and the unimodality of the h-vector of the determinantal ideal in these cases. We conjecture this result to be true for all values of k