Toric Ideals of Phylogenetic Invariants for the General Group-Based Model on Claw Trees K 1, n

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 Z2 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 basis ideal. In addition, we describe explicitly a quadratic lexicographic Gröbner 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 Z2 has a quadratic Gröbner basis. Hence, the coordinate ring of the toric variety is a Koszul algebra. Acknowledgment. The authors would like to thank Uwe Nagel for introducing us to the field of phylogenetic algebraic geometry and for his continuous support, motivation and guidance.