Toric ideals of phylogenetic invariants for the general group-based model on claw trees $K_{1,n}$

Toric ideals of phylogenetic invariants

Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the polynomials which vanish on the variety. Several widely used models for biological sequences have transition matrices that can be diagonalized by means of the Fourier transform of an abelian group. Their phylogenetic invariants form a toric ideal in the Fourier coordinates. We determine generators and Gröbner bases for these toric ideals. For the Jukes-Cantor and Kimura models on a binary tree, our…

Toric ideals of homogeneous phylogenetic models

We consider the phylogenetic tree model in which every node of the tree is observed and binary and the transitions are given by the same matrix on each edge of the tree. We are able to compute the Grobner basis and Markov basis of the toric ideal of invariants for trees with up to 11 nodes. These are perhaps the first non-trivial Grobner bases calculations in 2^11 indeterminates. We conjecture that there is a quadratic Grobner basis for binary trees. Finally, we give a explicit description of the polytope associated to this toric ideal for an infinite family of binary trees and conjecture that…

Toric geometry of cuts and splits

Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial structure of the graph and the corresponding cut polytope to algebraic properties of the ideal. Cut ideals generalize toric ideals arising in phylogenetics and the study of contingency tables.

Presentations of Semigroup Algebras of Weighted Trees

We find presentations for subalgebras of invariants of the coordinate algebras of binary symmetric models of phylogenetic trees studied by Buczynska and Wisniewski in \cite{BW}. These algebras arise as toric degenerations of rings of global sections of weight varieties of the Grassmanian of two planes associated to the Plücker embedding, and as toric degenerations of rings of invariants of Cox-Nagata rings.

The Cavender-Farris-Neyman Model with a Molecular Clock

We give a combinatorial description of the toric ideal of invariants of the Cavender-Farris-Neyman model with a molecular clock (CFN-MC) on a rooted binary phylogenetic tree and prove results about the polytope associated to this toric ideal. Key results about the polyhedral structure include that the number of vertices of this polytope is a Fibonacci number, the facets of the polytope can be described using the combinatorial "cluster" structure of the underlying rooted tree, and the volume is equal to an Euler zig-zag number. The toric ideal of invariants of the CFN-MC model has a quadratic G…

orbifold边界条件中的仿射半群：单位权重扇区中的割模型、系统发育模型与分层模型，以及超越它们的加权构型

An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them

本文确定二维orbifold边界条件对应的仿射半群，复现相关环不变量，提出加权字母表胶合树等贡献，分析不同阶群对应的系统发育等模型性质。