A tensorial approach to the inversion of group-based phylogenetic models

TL;DR

This paper introduces a tensorial method for inverting group-based phylogenetic models, generalizing Hadamard conjugation, and emphasizing the role of group representation theory for abelian groups.

Contribution

It presents a new tensorial framework for the inversion of group-based models, connecting group theory and matrix operations in phylogenetics.

Findings

Establishes a one-to-one correspondence between pattern probabilities and edge parameters.

Generalizes Hadamard conjugation using tensorial methods.

Highlights the elementary nature of the approach through matrix multiplication.

Abstract

Using a tensorial approach, we show how to construct a one-one correspondence between pattern probabilities and edge parameters for any group-based model. This is a generalisation of the "Hadamard conjugation" and is equivalent to standard results that use Fourier analysis. In our derivation we focus on the connections to group representation theory and emphasize that the inversion is possible because, under their usual definition, group-based models are defined for abelian groups only. We also argue that our approach is elementary in the sense that it can be understood as simple matrix multiplication where matrices are rectangular and indexed by ordered-partitions of varying sizes.