We explore the relationship between secant ideals and initial ideals of , the ideal of the Grassmannian, . The -secant of is the ideal generated by the subpfaffians of a generic skew-symmetric matrix. It has been conjectured that for a weight vector in the tropical Grassmannian, the secant of the initial ideal of with respect to 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 . In the course of proving this result, we show that the ideal is always prime, thus giving a new class of prime initial ideals of the Pfaffian ideals.
"Initial ideals of Pfaffian ideals." J. Commut. Algebra 12 (1) 91 - 105, Spring 2020. https://doi.org/10.1216/jca.2020.12.91