On the graph labellings arising from phylogenetics

We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.     

Conformal blocks,Berenstein–Zelevinsky triangles,and group-based models

Work of Buczyńska, Wi?niewski, Sturmfels and Xu, and the second author has linked the group-based phylogenetic statistical model associated with the group \(\mathbb {Z}/2\mathbb {Z}\) with the Wess–Zumino–Witten (WZW) model of conformal field theory associated to \(\mathrm {SL}_2(\mathbb {C})\) . In this article we explain how this connection can be generalized to establish a relationship between the phylogenetic statistical model for the cyclic group \(\mathbb {Z}/m\mathbb {Z}\) and the WZW model for the special linear group \(\mathrm {SL}_m(\mathbb {C}).\) We use this relationship to also show how a combinatorial device from representation theory, the Berenstein–Zelevinsky triangle, corresponds to elements in the affine semigroup algebra of the \(\mathbb {Z}/3\mathbb {Z}\) phylogenetic statistical model.