Vol. 50, No. 2, 1974

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Classes of circulants over the p-adic and rational integers

Dennis Garbanati

Vol. 50 (1974), No. 2, 435–447

Let G = {g0,g,g2,,gq1} be a finite abelian group of order q where q is a prime. Let Zp and Z denote the p-adic and rational integers respectively. A circulant for G over Zp (or Z) is a q-square matrix A of the form A = Σi=0q1aiP(gi) where ai Zp (or Z) and P is the left regular representation of G, i.e., P(gi) is a q-square permutation matrix and P(gigj) = P(gi)P(gj). Let M and L be symmetric unimodular circulants for G over Zp (or Z). The circulants M 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 = ATLA where T denotes transposition. The central object of this paper is:

(i) to give computable criteria for determining whether or not two circulants for G over Zp 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.

Mathematical Subject Classification 2000
Primary: 15A15
Received: 11 October 1972
Published: 1 February 1974
Dennis Garbanati