On a Lie-theoretic approach to generalized doubly stochastic matrices and applications

Lie Markov models

“…In fact we have the isomorphism GL 1 (n, C) ∼ = A(n − 1, C) where A(n − 1, C) is the (complex) affine group (see for example Baker (2003)). This observation allows the general methods of Lie theory to be applied to understanding continuous-time Markov models; see Johnson (1985) and Mourad (2004) for general results and discussion, and Sumner et al (2008) for applications to phylogenetics.Summarizing, we have the following set inclusionsand Lie group hierarchy e LGMM < GL 1 (n, C) < GL(n, C).We define a Markov model M by taking M ⊆ M GMM as some well defined subset of the general Markov model. Similarly, a rate-matrix model e L is defined by taking L ⊆ L GMM as some well defined subset of rate-matrices drawn from the general rate-matrix model and taking the set of exponentials thereof (as in (1)).…”

Markov Invariants for Phylogenetic Rate Matrices Derived from Embedded Submodels

“…We have chosen to write Q in terms of the natural basis of column-sum zero 'stochastic generator matrices' {L α , L β } of the group GL 1 (2); the subgroup of the general linear group GL(2) of invertible 2 × 2 matrices together with the probabilitistic constraint of unit-column sums (Johnson, 1985;Mourad, 2004). This is relevant for considerations of multiplicative closure of models, which might arise in applications where different rate matrices are allowed on different parts of a phylogenetic tree; where potentially missing taxa may need to be inserted into edges; or where re-evaluations of phylogeny may require edge rearrangements.…”

Lie Algebra Solution of Population Models Based on Time-Inhomogeneous Markov Chains

“…The idea of combining Lie algebras and symmetry considerations with Markov chains has continued to attract theoretical interest in a variety of contexts [6], [9], [12], [15].…”