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

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

2007-02-13

arxiv.org/abs/math/0702368v3

Mathematics

Commutative Algebra

Revised introduction (more complete); revised algorithms; added references. To appear in Proceedings of Algebraic Biology 2007

Scientific paper

We address the problem of studying the toric ideals of phylogenetic invariants for a general group-based model on an arbitrary claw tree. We focus on the group $\mathbb Z_2$ and choose a natural recursive approach that extends to other groups. The study of the lattice associated with each phylogenetic ideal produces a list of circuits that generate the corresponding lattice ideal. In addition, we describe explicitly a quadratic lexicographic Gr\"{o}bner basis of the toric ideal of invariants for the claw tree on an arbitrary number of leaves. Combined with a result of Sturmfels and Sullivant, this implies that the phylogenetic ideal of every tree for the group $\mathbb Z_2$ has a quadratic Gr\"{o}bner basis. Hence, the coordinate ring of the toric variety is a Koszul algebra.

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