Remarks on Hadamard conjugation and combinatorial phylogenetics

Hadamard conjugation can be used in the reconstruction of evolutionary trees and analysis of molecular sequence evolution. Despite the number of advantages Hadamard conjugation provides the number of evolutionary substitution models that can be used with this technique is limited. In this paper, we consider the question of whether Hadamard conjugation is limited to group-based evolutionary models, both for nucleotide substitution models and for codon substitution models. We expand the number of nucleotide substitution models that can be used with Hadamard conjugation and suggest new connections between phylogenetics and algebraic combinatorics especially with (commutative) association schemes.