Type-B generalized triangulations and determinantal ideals

Details Type-B generalized triangulations and determinantal ideals Type-B generalized triangulations and determinantal ideals

2006-07-06

arxiv.org/abs/math/0607159v1

Mathematics

Combinatorics

Scientific paper

For $n\geq 3$, let $\Omega_n$ be the set of line segments between the vertices of a convex $n$-gon. For $j\geq 2$, 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 $X_{ij}, 1\leq i,j\leq n$, 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 Gr\"obner-Basis whenever $k\in\{1,n-2,n-1\}$. We conjecture this result to be true for all values of $k

Affiliated with

Also associated with

No associations

Rating

Type-B generalized triangulations and determinantal ideals does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Type-B generalized triangulations and determinantal ideals, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Type-B generalized triangulations and determinantal ideals will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-45529