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

Work of Buczyńska, Wiśniewski, 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 SL₂(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 triangle, corresponds to elements in the affine semigroup algebra of the ℤ/3ℤ phylogenetic statistical model.