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Spring 2020 Initial ideals of Pfaffian ideals
Colby Long
J. Commut. Algebra 12(1): 91-105 (Spring 2020). DOI: 10.1216/jca.2020.12.91

Abstract

We explore the relationship between secant ideals and initial ideals of I2,n, the ideal of the Grassmannian, Gr(2,n). The (r1)-secant of I2,n is the ideal generated by the 2r×2r subpfaffians of a generic n×n skew-symmetric matrix. It has been conjectured that for a weight vector ω in the tropical Grassmannian, the secant of the initial ideal of I2,n with respect to ω 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 ω. In the course of proving this result, we show that the ideal inω(I2,n{r}) is always prime, thus giving a new class of prime initial ideals of the Pfaffian ideals.

Citation

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Colby Long. "Initial ideals of Pfaffian ideals." J. Commut. Algebra 12 (1) 91 - 105, Spring 2020. https://doi.org/10.1216/jca.2020.12.91

Information

Received: 26 October 2016; Revised: 3 May 2017; Accepted: 11 May 2017; Published: Spring 2020
First available in Project Euclid: 13 May 2020

zbMATH: 07211328
MathSciNet: MR4097059
Digital Object Identifier: 10.1216/jca.2020.12.91

Subjects:
Primary: 14M25

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.12 • No. 1 • Spring 2020
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