Systematics and symmetry in molecular phylogenetic modelling

Probabilistic models of mathematical phylogenetics have been intensively used in recent years in biology as the cornerstone of methods to infer and reconstruct the ancestral relationships between species. We bring a lens of mathematical physics to bear on the formulation of the theoretical models, focussing on the applicability of group and representation theory to guide model specification and to exploit the multilinear setting of the models in the presence of underlying symmetries. We focus on aspects of multipartite entanglement which are shared between descriptions of quantum states on the physics side, and the multi-way tensor probability arrays arising in phylogenetics. We give examples of entanglement invariants (Markov invariants) for the case of quartets, and DNA data; and compare and contrast the rings of quantum/Markov invariants for the three and four qubit case/binary triplet and quartet cases, respectively.