Pseudo-direct sums and wreath products of loose-coherent algebras with applications to coherent configurations

We introduce the notion of a loose-coherent algebra, which is a special semisimple subalgebra of the matrix algebra, and define two operations to obtain new loose-coherent algebras from the old ones: the pseudo-direct sum and the wreath product. For two arbitrary coherent configurations C, D and their wreath product C≀D, it is difficult to express the Terwilliger algebra T(x,y)(C≀D) in terms of the Terwilliger algebras Tx(C) and Ty(D). By using the concept and operations of loose-coherent algebras, we find a very simple such expression. As a direct consequence of this expression, we obtain the central primitive idempotents of T(x,y)(C≀D) in terms of the central primitive idempotents of Tx(C) and Ty(D). Many results in [4], [6], [10], [12] are special cases of the results in this paper.