Type-B generalized triangulations and determinantal ideals

Abstract

For $n\ge 3$, let ${\Omega }_n$ be the set of line segments between the vertices of a convex $n$-gon. For $j\ge 2$, a $j$-crossing is a set of $j$ line segments pairwise intersecting in the relative interior of the $n$-gon. We identify line-segments in ${\Omega }_{2n}$ which can be transformed into each other by a $180^{\circ }$-rotation of the $2n$-gon. Let $\F_n$ be the set ${\Omega }_{2n}$ 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. We demonstrate that $\D_{n,k}$ 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. We give a term-order on the monomials in the variables $X_{ij},1\le i,j\le n$, such that the corresponding initial ideal of the determinantal ideal generated by the $(k+1)$ times $(k+1)$ minors of the generic $n\times n$ matrix contains the Stanley-Reisner ideal of $\D_{n,k}$. We show that the minors form a Gröbner-Basis whenever $k\in \{1,n-2,n-1\}$. We conjecture this result to be true for all values of $k<n$.

References in corpus (1)

• Realizations of the associahedron and cyclohedron