  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.
