2026/08/06 by Yue Meng, Zhiqiang Yu
Mathematics · #math.QA #math.CT #msc:18M20
21 pages; comments are welcome
arxiv created 2026/08/06 · arxiv updated 2026/08/07
A fusion category C is said to be ℤ/2ℤ× ℤ/2ℤ-quadratic if the group G(C) of invertible objects is isomorphic to ℤ/2ℤ× ℤ/2ℤ, and the remaining simple objects form an orbit under the action of G(C). In this paper, we give a partial classification of ℤ/2ℤ× ℤ/2ℤ-quadratic fusion categories of rank six. More precisely, we show that its Grothendieck ring K0(C) must be one of nine fusion rings if the fusion rule multiplicities are less than 20, and the categorifications of five of them are previously known. We prove that one of the last four fusion rings can be realized as de-equivariantization of a near-group fusion category of type ℤ/2ℤ× ℤ/4ℤ+8.