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.