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

Abstract

Work of Buczynska, Wiśniewski, Sturmfels and Xu, and the second author has linked the group-based phylogenetic statistical model associated with the group $$\mathbb {Z}/2\mathbb {Z}$$ Z / 2 Z with the Wess---Zumino---Witten (WZW) model of conformal field theory associated to $$\mathrm {SL}_2(\mathbb {C})$$ 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 $$\mathbb {Z}/m\mathbb {Z}$$ Z / m Z and the WZW model for the special linear group $$\mathrm {SL}_m(\mathbb {C}).$$ 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 $$\mathbb {Z}/3\mathbb {Z}$$ Z / 3 Z phylogenetic statistical model.