On p-standard table algebras of order p3

Abstract

The p-standard table algebras of order p3 for any rational prime p, which include the adjacency algebras of p-schemes of order p3, are classified, provided that the thin radical of the distinguished basis of the table algebra has order p. The classification is in terms of wreath products, and of certain vectors (that we call Legendre elements) in the group algebra of a group of order p. Their coefficients yield real solutions to the classical equations that hold for the Legendre symbols. A fully explicit classification of the relevant Legendre elements, and hence of the table algebras at issue, is attained in the case of integer structure constants, as we characterize certain real solutions to these equations sufficiently enough to show that the only integer solutions are in fact ± the Legendre symbols themselves and appropriate translates thereof. Criteria for two such table algebras being isomorphic are also established in this case, and the automorphism group of any one such table algebra is determined.