Fusion rings with few degrees

The main result characterizes fusion rings whose degree set consists of 1 and a fixed prime p. A well known theorem of Isaacs and Passman on finite groups whose irreducible character degrees are 1 and p follows as a corollary. Numerically compatible (NC) fusion rings are defined as a potential generalization of the Grothendieck rings of fusion categories; and the NC fusion rings of order p, p(2), p(3), pq, and pq(2) (where p, q are primes) are classified in terms of certain standard table algebras, as another application of the main theorem. (C) 2013 Elsevier Inc. All rights reserved.