Genome rearrangements as double coset Markov chains

Permutation groups provide a natural framework for modeling genomes and their rearrangement events, which is crucial for understanding evolutionary relationships. Double cosets can be used to model objects arising from multiple symmetries, such as circular genomes with repeated genes. Suppose λ = ( λ ₁,λ ₂, , λ ₖ) λ = ( λ 1 , λ 2 , ⋯ , λ k ) is a partition of n n which indicates the number of repeated regions or genes of different types ( λ ᵢ λ i of type i i with n n total regions). A circular genome with n n oriented regions determined by λ λ can be identified with the double coset space S_λ B_n / D_n S λ \ B n / D n , where Bₙ B n is the hyperoctahedral group, Dₙ D n the dihedral group extended to Bₙ B n , and S_λ S λ