On the ideals of equivariant tree models

Abstract

We introduce equivariant tree models in algebraic statistics, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group-based models such as the Jukes-Cantor and Kimura models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices. The main novelty is that our results yield generators of the full ideal rather than an ideal which only defines the model set-theoretically. Set-up and theoremsIn phylogenetics, tree models have been introduced to describe the evolution of a number of species from a distant common ancestor. Given suitably aligned strings of nucleotides of n species alive today, one assumes that the individual positions in J. Draisma has been supported by DIAMANT, an NWO mathematics cluster and J. Kuttler by an NSERC Discovery Grant.

Lectures on Algebraic Statistics

“…This was conjectured in [50, §7] and proved in [5] and in [65]. The DraismaKuttler theorem [36] can be regarded as a generalization of this statement.…”

“…We here do not offer a formal definition of what "reasonable" means but refer to [36] instead. Let K 1,m denote the complete bipartite graph with one hidden node and m observed nodes.…”

Finite phylogenetic complexity ofZpand invariants forZ3Michałek1 2017European Journal of Combinatorics2010View full textAdd to dashboardCite

“…For very general, so-called equivariant models, the fact that on set-theoretic level there exists a bound was proved in [DK09,DK14,DE15]. For the class of G-models that includes all the models introduced in this article, on the level of projective schemes the bounds were obtained in [Mic13].…”

“…These trees, denoted by K 1,n have one inner vertex and n leaves. Let us cite Draisma and Kuttler [DK09]:…”

Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals

“…1. For other recent results about Sym(N)-Noetherian polynomial algebras see, for instance, articles [2,8,10,11,12] and a survey [7]. 2.…”