Algebraic varieties representing group-based Markov processes on trees

TL;DR

This paper advances the algebraic understanding of group-based Markov models on trees by proving pseudo-toricity and normality of associated varieties, and extends algorithms for their analysis using Fourier transforms, sockets, and networks.

Contribution

It generalizes previous results by Sullivant and Sturmfels, providing new conditions for normality and a broader algorithmic framework for algebraic models on trees.

Findings

Many models are pseudo-toric

Identifies cases where varieties are normal

Develops a generalized Fourier transform approach

Abstract

In this paper we complete the results of Sullivant and Sturmfels proving that many of the algebraic group-based models for Markov processes on trees are pseudo-toric. We also show in which cases these varieties are normal. This is done by the generalization of the discrete Fourier transform approach. In the next step, following Sullivant and Sturmfels, we describe a fast algorithm finding a polytope associated to these algebraic models. However in our case we apply the notions of sockets and networks extending the work of Buczynska and Wisniewski who introduced it for the binary case.