Initial ideals of Pfaffian ideals

Abstract

We explore the relationship between secant ideals and initial ideals of $I_{2,n}$, the ideal of the Grassmannian, $Gr\left(2,{ℂ}^n\right)$. The $\left(r-1\right)$-secant of $I_{2,n}$ is the ideal generated by the $2r\times 2r$ subpfaffians of a generic $n\times n$ skew-symmetric matrix. It has been conjectured that for a weight vector $\omega$ in the tropical Grassmannian, the secant of the initial ideal of $I_{2,n}$ with respect to $\omega$ is equal to the initial ideal of the secant. We show that this conjecture is not true in general. Using the correspondence between weight vectors in the tropical Grassmannian and binary leaf-labeled trees, we also give necessary and sufficient conditions for the conjecture to hold in terms of the topology of the tree associated to $\omega$. In the course of proving this result, we show that the ideal ${in}_{\omega }\left(I_{2,n}^{\left\{r\right\}}\right)$ is always prime, thus giving a new class of prime initial ideals of the Pfaffian ideals.