On fusion categories with few irreducible degrees

SONIA NATALE; JULIA YAEL PLAVNIK

Mendoza

Congreso; Coloquio de Álgebra y Teoría de Representaciones; 2012

UNC - UNCuyo

In this talk we shall consider the general problem of understanding the structure of a fusion category C after the knowledge of the set cd(C) of Frobenius- Perron dimensions of its simple objects. For a finite group G, the knowledge of the set cd(G) = cd(kG) gives in some cases substantial information about the structure of G. It is known, for instance, that if |cd(G)| is at most 3, then G is solvable. In particular, we consider the case where cd(C) = {1, p}, with p a prime number. We shall show various structural results regarding nilpotency and solvability, in the sense introduced by Etingof, Gelaki, Nikshych and Ostrik. The talk is based on joint work with S. Natale (preprint: arXiv:1103.2340v2).