Classes of circulants over thep-adic and rational integers
Dennis A. Garbanati · Pacific Journal of Mathematics · 1974
1 } be a finite abelian group of order q where q is a prime.Let Z p and Z denote the p-adic and rational integers respectively.A circulant for G over Z p (or Z) is a g-square matrix A of the form A = 2?=o ^iP(9*) where α* e Z p (or Z) and P is the left regular representation of G, i.e., P(flf*) is a g-square permutation matrix and P{g i g j ) = Pig^Pigt).Let ikf and L be symmetric unimodular circulants for G over Z p (or Z).The circulants ikf and L are said to be in the same G-class if there exists a circulant A for G over Zp (or Z, respectively) such that M-A τ LA where τ denotes transposition.The central object of this paper is: (i) to give computable criteria for determining whether or not two circulants for G over Z p are in the same G-class, (ii) to give a computable upper bound (which seems to be frequently equal to 1) for the number of G-classes among the positive definite symmetric unimodular circulants, and (iii) to introduce a group matrix concept (called G-genus) corresponding to the concept of genus.