The structure of Schur rings over cyclic groups

Abstract

Let G be a finite group and D1,D2,...,Dd be a partition of G. Suppose, for each i=1,2,...,d, {gε{lunate}G{divides}g-1ε{lunate}Di};=D j for some j depending on i; and D ̄i D ̄j=∑d h=1 Ph ij D ̄h for all i,j=1,2,...,d where D ̄m=∑gε{lunate}Dmgε{l unate}C[G]. Then the subalgebra of C[G] spanned by D̄1,D̄ 2,...,D̄d is called a Schur 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 when G is a cyclic group. Two special cases are thoroughly determined. The first one concerns with the case that every Di is fixed by Aut G. For the second one, we consider the case that G is a cyclic p-group. Also, examples of Schur rings with low dimension are given. © 1990.