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Let G = {g0,g,g2,⋯,gq−1}
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=0q−1aiP(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.
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