Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algebras

Let (G, Ω) be a permutation group of degree n. Let V(G, Ω) be the set of all square matrices of order n which commute with all permutation matrices corresponding to permutations from (G, Ω). V(G, Ω) is a matrix algebra which is called the centralizer algebra of (G, Ω). In this paper we introduce the combinatorial analogue of centralizer algebras, namely coherent (cellular) algebras and consider the properties of these algebras. It turns out that coherent algebras provide a very helpful tool for the investigation of the symmetries of graphs of different kinds, in particular, of molecular graphs.