Combinatorial extensions of Terwilliger algebras and wreath products of association schemes

We introduce the notion of the combinatorial extension of a Terwilliger algebra by a coherent algebra. By using this notion, we find a simple way to describe the Terwilliger algebras of certain coherent configurations as combinatorial extensions of simpler Terwilliger algebras. In particular, given an association scheme S and another association scheme R such that the Terwilliger algebra of R is isomorphic to a coherent algebra, we prove that the Terwilliger algebra of the wreath product S≀R is isomorphic to the combinatorial extension of the Terwilliger algebra of S by a coherent algebra. We also show that the Terwilliger algebra of the wreath product W of rank 2 association schemes can be expressed as the combinatorial extension of adjacency algebras of association schemes induced by the closed subsets of W. As a direct consequence, we obtain simple conceptual explanations and alternative proofs of many known results on the structures of Terwilliger algebras of wreath products of association schemes.