Permutation group approach to association schemes

AbstractWe survey the modern theory of schemes (coherent configurations). The main attention is paid to the schurity problem and the separability problem. Several applications of schemes to constructing polynomial-time algorithms, in particular, graph isomorphism tests, are discussed

Terwilliger algebras of wreath products by quasi-thin schemes

The structure of Terwilliger algebras of wreath products by thin schemes or one-class schemes was studied in [A. Hanaki, K. Kim, Y. Maekawa, Terwilliger algebras of direct and wreath products of association schemes, J. Algebra 343 (2011) 195--200]. In this paper, we will consider the structure of Terwilliger algebras of wreath products by quasi-thin schemes. This gives a generalization of their result

Characterization of Balanced Coherent Configurations

Let $G$ be a group acting on a finite set $\Omega$. Then $G$ acts on $\Omega\times \Omega$ by its entry-wise action and its orbits form the basis relations of a coherent configuration (or shortly scheme). Our concern is to consider what follows from the assumption that the number of orbits of $G$ on $\Omega_i\times \Omega_j$ is constant whenever $\Omega_i$ and $\Omega_j$ are orbits of $G$ on $\Omega$. One can conclude from the assumption that the actions of $G$ on ${\Omega_i}$'s have the same permutation character and are not necessarily equivalent. From this viewpoint one may ask how many inequivalent actions of a given group with the same permutation character there exist. In this article we will approach to this question by a purely combinatorial method in terms of schemes and investigate the following topics: (i) balanced schemes and their central primitive idempotents, (ii) characterization of reduced balanced schemes