On Circulant Matrices

S ome mathematical topics—circulant matrices, in particular—are pure gems that cry out to be admired and studied with different techniques or perspectives in mind. Our work on this subject was originally motivated by the apparent need of the first author to derive a specific result, in the spirit of Proposition 24, to be applied in his investigation of theta constant identities [9]. Although progress on that front eliminated the need for such a theorem, the search for it continued and was stimulated by enlightening conversations with Yum-Tong Siu during a visit to Vietnam. Upon the first author’s return to the U.S., a visit by Paul Fuhrmann brought to his attention a vast literature on the subject, including the monograph [4]. Conversations in the Stony Brook mathematics common room attracted the attention of the second author and that of Sorin Popescu and Daryl Geller∗ to the subject and made it apparent that circulant matrices are worth studying in their own right, in part because of the rich literature on the subject connecting it to diverse parts of mathematics. These productive interchanges between the participants resulted in [5], the basis for this article. After that version of the paper lay dormant for a number of