On association schemes of finite exponent

Our purpose is to search association schemes which are obtained by repeating an extension by a thin normal closed subset. To do this, we will focus the higher Frobenius-Schur indicators which are studied in many research areas recently. The classical Frobenius-Schur indicator for association schemes was introduced by Higman [3]. We will define the higher Frobenius-Schur indicators for association schemes as a generalization of this indicator in §3. We will make a conjecture on the connection between the values of the higher Frobenius-Schur indicators of the regular representation and association schemes obtained by repeating an extension by a thin normal closed subset. In §4, we will define the girth and the strong girth of a relation of an association scheme as candidates correspond to the order of an element of a finite group. Moreover, we will define the exponent of an association scheme by the least common multiple of the strong girths of all relations. We will show that our conjecture holds for a class of association schemes of finite exponent.