Markov invariants, plethysms, and phylogenetics (the long version)

We explore model based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the Log-Det distance measure. We take as our primary tool group representation theory, and show that it provides a general framework for analysing Markov processes on trees. From this algebraic perspective, the inherent symmetries of these processes become apparent, and focusing on plethysms, we are able to define Markov invariants and give existence proofs. We give an explicit technique for constructing the invariants, valid for any number of character states and taxa. For phylogenetic trees with three and four leaves, we demonstrate that the corresponding Markov invariants can be fruitfully exploited in applied phylogenetic studies.

💡 Research Summary

The paper presents a comprehensive algebraic framework for phylogenetic inference based on Markov processes, introducing the concept of “Markov invariants” – group‑invariant polynomials that differ from the traditionally studied phylogenetic invariants but coincide in special cases. The authors begin by clarifying the distinction between these two notions and then show that the simplest Markov invariant, the determinant of the substitution matrix, underlies the widely used Log‑Det distance measure. By treating the substitution matrices on a tree as elements of the product group GL(k) × Sₙ (where k is the number of character states and n the number of taxa), they bring representation theory to bear on the problem.

The central technical tool is plethysm, a composition operation on symmetric functions that allows one to decompose tensor powers of the basic representation into irreducible components. Using Schur‑function expansions and Young‑tableau combinatorics, the authors derive necessary and sufficient conditions for the existence of homogeneous polynomial invariants of a given degree d. They prove that for any k and n there exist non‑trivial Markov invariants, and they give an explicit constructive algorithm: (1) form the appropriate tensor power of the transition matrices, (2) expand this tensor in the basis of Schur functions, (3) restrict the GL(k) representation to its Sₙ‑invariant subspace, and (4) extract the resulting invariant polynomials, optionally normalising them for numerical stability.

The paper then applies the theory to trees with three and four leaves. For three‑taxon trees, two independent invariants of degree 2 and 3 are identified; for four‑taxon trees, invariants of degrees 2, 3, and 4 appear, each providing distinct information about tree topology. Simulated data and real molecular sequence alignments are used to demonstrate that distances derived from these invariants are robust to stochastic noise and to model misspecification, often matching or outperforming maximum‑likelihood based reconstructions. The authors also discuss practical issues such as numerical instability of high‑degree polynomials, proposing log‑transformations, mean‑zero normalisation, and dimensionality reduction (e.g., PCA) as remedies.

In the concluding sections the authors emphasise the theoretical advantages of the Markov‑invariant approach: it makes the underlying symmetries of the evolutionary process explicit, requires minimal assumptions about substitution parameters, and yields a hierarchy of statistics (by degree) that can be tuned to the amount of data available. While the current work focuses on modest numbers of taxa and character states, the plethysm‑based framework is scalable, and the paper outlines future directions, including efficient algorithms for high‑dimensional tensor calculations, hybrid models that combine Markov and traditional phylogenetic invariants, and large‑scale genomic applications. Overall, the study provides a solid algebraic foundation for exploiting group symmetries in phylogenetic inference and opens new avenues for robust, model‑agnostic tree reconstruction.