Phylogenetic invariants for group-based models (Q2819992)

In phylogenetics, possible histories of evolution are represented by trees, and to each tree can be associated a certain algebraic variety. Phylogenetic invariants are polynomials that vanish on this algebraic variety and are used to distinguish between phylogenetic trees. The purpose of this article is to propose a new method for computing phylogenetic invariants of general group-based models. Group-based models are a family of models for which the associated algebraic varieties are known to be toric [\textit{S. N. Evans} and \textit{T. P. Speed}, Ann. Stat. 21, No. 1, 355--377 (1993; Zbl 0772.92012); \textit{L. A. Székely} et al., Adv. Appl. Math. 14, No. 2, 200--216 (1993; Zbl 0794.05014); \textit{M. Michałek}, J. Algebra 339, No. 1, 339--356 (2011; Zbl 1251.14040)].NEWLINENEWLINEFrom the introduction, the main conjecture of the article (Conjecture 12) proposes that ``the varieties associated to large claw trees are scheme-theoretic intersections of varieties associated to trees of smaller valency.'' This would imply that phylogenetic invariants of a claw tree could be computed recursively. This conjecture is important because work of Sullivant and Sturmfels yields a description of phylogenetic invariants of a tree if the invariants of claw trees are known [\textit{B. Sturmfels} and \textit{S. Sullivant}, J. Comp. Biol. 12, 204--228 (2005; Zbl 1391.13058)]. The authors prove their conjecture for the binary Jukes-Cantor model and show that it is equivalent to a conjecture of Sullivant and Sturmfels for the 3-Kimura model. They also give some new examples of non-normal varieties associated to general group-based models for an abelian group.