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

Work of BuczyA"ska, WiA > niewski, Sturmfels and Xu, and the second author has linked the group-based phylogenetic statistical model associated with the group with the Wess-Zumino-Witten (WZW) model of conformal field theory associated to . In this article we explain how this connection can be generalized to establish a relationship between the phylogenetic statistical model for the cyclic group and the WZW model for the special linear group 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 phylogenetic statistical model.