Tables of practical invariants for distinguishing multiplicity-free fusion categories up to rank 7

TL;DR

This paper provides tables of invariants for multiplicity-free fusion categories up to rank 7, aiding their classification and comparison, and discusses the correctness and completeness of existing categorizations.

Contribution

It introduces practical invariants for distinguishing fusion categories, enabling manual verification of automorphisms and inequivalence, and assesses the classification's accuracy up to rank 7.

Findings

Invariants can be checked manually for automorphisms.

Invariants can determine inequivalence of categories.

The classification up to rank 7 is consistent under correct data.

Abstract

This paper discusses to what extent the census of multiplicity-free fusion categories up to rank 7, proposed by the software package Anyonica and the anyonwiki website, can be regarded as a proper classification. The questions of correctness and completeness are briefly discussed, and the question of inequivalence of the provided categories is resolved under the assumption that all proposed data is correct. This is done by providing tables of small sets of invariants for these categories, for which (a) their invariance under automorphisms can be checked manually, and (b) the (in)equivalence of two skeletal fusion categories can be checked manually. These invariants can also be used to identify the category on the anyonwiki that corresponds to one for which the skeletal data is known.