Schur rings over dihedral groups

Let G be a finite group and P = {D₀ = {e}, D₁, · · · , Dd} be a partition of G. Suppose, for each i, j, 0 ≤ i, j ≤ d,{g ∊ G|g⁻¹ ∊ Di} = Di* ∊ P for some 0 ≤ i* ≤ d and ${\bar D_i}{\bar D_j} = \sum\limits_{k = 0}^d {p_{ij}^k} {\bar D_k}where{\bar D_m} = \sum\limits_{g \in {D_m}} {g \in \mathbb{C}\left[ G \right]} $. Then the subalgebra of ℂ[G] spanned by D̅₀, · · · , D̅d is called a Schur ring (S-ring). Such an object is known to have application on group theory and combinatorial design theory. In this paper, we study the structure of Schur rings over dihedral group Dn, Special attention is paid to the case when n = p where p is an odd prime.