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

TL;DR

This paper generalizes the connection between group-based phylogenetic models and conformal field theory from Z/2Z to Z/mZ, linking combinatorial representation theory tools to algebraic structures in phylogenetics.

Contribution

It extends the known relationship between Z/2Z models and WZW theory to Z/mZ, and relates Berenstein-Zelevinsky triangles to algebraic elements in the model.

Findings

Established a correspondence between Z/mZ models and SL(m,C) WZW models.

Connected Berenstein-Zelevinsky triangles to affine semigroup algebra elements.

Generalized previous Z/2Z results to cyclic groups Z/mZ.

Abstract

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.