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

Work of Buczynska, Wisniewski, Sturmfels and Xu, and the second author has linked the group-based phylogenetic statistical model associated with the group Z/2Z with the Wess-Zumino-Witten (WZW) model of conformal field theory associated to SL(2,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 Z/mZ and the WZW model for the special linear group SL(m,C). We use this relationship to also show how a combinatorial device from representation theory, the Berenstein-Zelevinsky triangles, correspond to elements in the affine semigroup algebra of the Z/3Z phylogenetic statistical model.