Initial Ideals of Pfaffian Ideals

TL;DR

This paper proves a conjecture about binomial initial ideals of Grassmannian ideals related to phylogenetic trees, establishing conditions for secant ideals and introducing new prime initial ideals of Pfaffian ideals.

Contribution

It resolves a conjecture on binomial initial ideals associated with phylogenetic trees and characterizes when secant ideals coincide, also introducing new prime initial ideals of Pfaffian ideals.

Findings

Necessary and sufficient conditions for secant ideal equality.

Identification of prime initial ideals of Pfaffian ideals.

Connection between tropical Grassmannian weight vectors and tree topologies.

Abstract

We resolve a conjecture about a class of binomial initial ideals of $I_{2,n}$, the ideal of the Grassmannian, Gr$(2,\mathbb{C}^n$), which are associated to phylogenetic trees. For a weight vector $\omega$ in the tropical Grassmannian, $in_\omega(I_{2,n}) = J_\mathcal{T}$ is the ideal associated to the tree $\mathcal{T}$. The ideal generated by the $2r \times 2r$ subpfaffians of a generic $n \times n$ skew-symmetric matrix is precisely $I_{2,n}^{\{r-1\}}$, the $(r-1)$-secant of $I_{2,n}$. We prove necessary and sufficient conditions on the topology of $\mathcal{T}$ in order for $in_\omega(I_{2,n})^{\{2\}} = J_\mathcal{T}^{\{2\}}$. We also give a new classof prime initial ideals of the Pfaffian ideals.