Point configurations, phylogenetic trees, and dissimilarity vectors

Author:

Caminata AlessioORCID,Giansiracusa NoahORCID,Moon Han-BomORCID,Schaffler LucaORCID

Abstract

In 2004, Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter and Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors.

Funder

National Science Foundation

Publisher

Proceedings of the National Academy of Sciences

Subject

Multidisciplinary

Reference22 articles.

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2. Reconstructing trees from subtree weights

3. On the relation between weighted trees and tropical Grassmannians;Cools;J. Symbolic Comput.,2009

4. Dissimilarity vectors of trees are contained in the tropical Grassmannian;Giraldo;Electron. J. Comb.,2010

5. Dissimilarity maps on trees and the representation theory of S L m ( C );Manon;J. Algebr. Comb.,2011

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Determinantal Varieties From Point Configurations on Hypersurfaces;International Mathematics Research Notices;2023-10-19

2. A Pascal's theorem for rational normal curves;Bulletin of the London Mathematical Society;2021-06-15

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