Schur rings and cyclic association schemes of class three

Abstract

LetG be a group of finite order andD 0 = {e},D 1,...,D d be a partition ofG. Suppose{d ?1|d ?D i } =D i?, i? ? {0, 1,..., d}, for eachi ? {0, 1,..., d}; and $$\bar D_i ,\bar D_j = \sum\limits_{k = 0}^d {p_{ij}^k } \bar D_k $$ for alli, j where $$\bar D_m , = \sum\limits_{g \in D_m } g \in \mathbb{C}[G]$$ . Then the subalgebra spanned by $$\bar D_0 ,\bar D_1 , \ldots ,\bar D_d $$ is called a Schur ring overG. It is known that such a partitionD 0,D 1,...,D d can be used to construct an association scheme of classd. In this paper, we obtain a complete classification for the case whenG is cyclic andd = 3. The result corresponds to a complete classification of cyclic association schemes of class three.