Necessary and sufficient conditions for the existence of linear invariants in phylogenetic inference

The necessary and sufficient conditions for the existence of linear invariants under semigroups of probability transition matrices are derived. It is found that a biologically meaningful nucleotide substitution model has linear invariants if and only if it is a submodel of one of the three most general models, which include the so-called balanced and unbalanced transversion models. Each of these three general models is a nucleotide substitution model with six parameters.