Flatness and fusion coefficients

Abstract

This paper addresses the problem of how to directly read off the 'fusion coefficients' (for the tensor-products of bimodules arising in the 'tower of the basic construction) for a subfactor, from the data of the associated paragroup. The solution to this problem is closely related to a reformulation, which is obtained here, of the flatness condition to be satisfied by the connection on a paragroup; our reformulation is best described by the phrase '1 ∗ flatness', in contrast with Ocneanu's initial formulation as '4 ∗ flatness' and Kawahigashi's subsequent reformulation as '2 ∗ flatness'.