Type-B generalized triangulations and determinantal ideals

Abstract

For n ≥ 3 , let Ω n 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 Δ n , k be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of Ω n not containing any ( k + 1 ) -crossing. The complex Δ n , 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 Ω 2 n which can be transformed into each other by a 180 ∘-rotation of the 2 n -gon. Let F n be the set Ω 2 n after identification, then the complex D n , k of type-B generalized triangulations is the simplicial complex of subsets of F n not containing any ( k + 1 ) -crossing in the above sense. For k = 1 , we have that D n , 1 is the simplicial complex of type-B triangulations of the 2 n -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 D n , k is a pure, k ( n − k ) − 1 + k n dimensional complex that decomposes into a k n − 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 D n , k . On the algebraical side we give a term order on the monomials in the variables X i j , 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 D n , 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 < n .