Initial Ideals of Pfaffian Ideals

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, $i{n}_{\omega }(I_{2,n})=J_{\mathcal{T}}$ 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 $i{n}_{\omega }(I_{2,n}{)}^{\{2\}}=J_{\mathcal{T}}^{\{2\}}$. We also give a new classof prime initial ideals of the Pfaffian ideals.

14 pages