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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


We explore the relationship between secant ideals and initial ideals of I 2 , n , the ideal of the Grassmannian, Gr ( 2 , n ) . The ( r 1 ) -secant of I 2 , n is the ideal generated by the 2 r × 2 r 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 I 2 , 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 in ω ( I 2 , n { r } ) is always prime, thus giving a new class of prime initial ideals of the Pfaffian ideals.


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Colby Long. "Initial ideals of Pfaffian ideals." J. Commut. Algebra 12 (1) 91 - 105, Spring 2020.


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

Primary: 14M25

Keywords: initial ideal , Pfaffian ideal , Phylogenetics , secant ideal

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium


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Vol.12 • No. 1 • Spring 2020
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