Narrow Results

We eliminate 39 infinite families of possible principal graphs as part of the classification of subfactors up to index 5. A number-theoretic result of Calegari–Morrison–Snyder, generalizing Asaeda–Yasuda, reduces each infinite family to a finite number of cases. We provide algorithms for computing the effective constants that are required for this result, and we obtain 28 possible principal graphs. The Ostrik d-number test and an algebraic integer test reduce this list to seven graphs in the index range (4,5) which actually occur as principal graphs.

Given a discrete group G and a spherical G-fusion category whose neutral component has invertible dimension, we use the state-sum method to construct a 3-dimensional Homotopy Quantum Field Theory with target the Eilenberg–MacLane space K(G, 1).

We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion, we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary re-proof of the fact that the quadruple dual functor is naturally isomorphic to the identity.

We exhibit two groups of order 16 that are categorically Morita equivalent, but not Grothendieck equivalent, and therefore not isocategorical. These groups are the smallest such examples.

In this paper, we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois group of their character tables. Based on these symmetries, we generalize a well-known result of Burnside from representation theory of finite groups. More precisely, we show that any row corresponding to a non-invertible object in the character table of a weakly integral fusion category contains a zero entry.