Combinatorial Extensions of Terwilliger Algebras and Wreath Products of Association Schemes
Mathematics and Statistics
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 SS and another association scheme RR such that the Terwilliger algebra of RR 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 SS by a coherent algebra. We also show that the Terwilliger algebra of the wreath product WW of rank 22 association schemes can be expressed as the combinatorial extension of adjacency algebras of association schemes induced by the closed subsets of WW. 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.
Song, S. Y., Xu, B., & Zhou, S. (2017). Combinatorial extensions of Terwilliger algebras and wreath products of association schemes. Discrete Mathematics, 340(5), 892-905. doi:10.1016/j.disc.2017.01.015