Classification of low-rank odd-dimensional modular categories (Q6565650)

The problem of classifying modular tensor categories (MTCs) is under active research. It was shown in [\textit{P. Bruillard} et al., J. Am. Math. Soc. 29, No. 3, 857--881 (2016; Zbl 1344.18008)] that there are finitely many MTCs of a fixed rank, up to equivalence. A classification of unitary MTCs of rank at most 4 was presented in [\textit{E. Rowell} et al., Commun. Math. Phys. 292, No. 2, 343--389 (2009; Zbl 1186.18005)]. All MTCs of rank at most 5 with some object being non-self-dual were classified in [\textit{S.-M. Hong} and \textit{E. Rowell}, J. Algebra 324, No. 5, 1000--1015 (2010; Zbl 1210.18006)]. Later on, all possible fusion rules for MTCs of rank 5 were determined in [\textit{P. Bruillard} et al., Int. Math. Res. Not. 2016, No. 24, 7546--7588 (2016; Zbl 1404.18016)], being used to describe their classification up to monoidal equivalence. Recently, a classification of rank 6 MTCs up to modular data were given in [\textit{S.-H. Ng} et al., Commun. Math. Phys. 402, No. 3, 2465--2545 (2023; Zbl 1519.18012)], while the classification of all integral MTCs of rank at most 12 was reported in [\textit{M. A. Alekseyev} et al., ``Classification of integral modular data up to rank 13'', Preprint, \url{arXiv:2302.01613}].\N\NIt was established in [\textit{P. Bruillard} and \textit{E. C. Rowell}, Proc. Am. Math. Soc. 140, No. 4, 1141--1150 (2012; Zbl 1262.18005)] that odd-dimensional MTCs of rank at most 11 are pointed, where an example of odd-dimensional MTC of rank 25 that is not pointed was given by the representation category \(\mathrm{Rep}\left( D^{\omega}\left( \mathbb{Z}_{7}\rtimes\mathbb{Z}_{3}\right) \right) \), and it was asked whether this is the smallest example of a non-pointed odd-dimensional MTC. A partial answer to this question was given in [\textit{A. Czenky} and \textit{J. Plavnik}, Algebra Number Theory 16, No. 8, 1919--1939 (2022; Zbl 1516.18016)], where it was shown that all odd-dimensional MTCs of rank 13 or 15 must be pointed, and all odd-dimensional MTCs of rank between 17 and 23 are either pointed or perfect.\N\NThis paper, continuing the study of low-rank odd-dimensional MTCs, shows that odd-dimensional MTCs of ranks 17 to 23 are all pointed, which settles the question in [\textit{P. Bruillard} and \textit{E. C. Rowell}, Proc. Am. Math. Soc. 140, No. 4, 1141--1150 (2012; Zbl 1262.18005)] affirmatively. Pointed MTCs are classified by pairs \(\left( G,q\right) \), where \(G\)\ is a finite abelian group, and \(q:G\rightarrow\boldsymbol{k}^{\times}\)\ is a non-degenerate quadratic form on \(G\)\ [\textit{P. Bruillard} and \textit{E. C. Rowell}, Proc. Am. Math. Soc. 140, No. 4, 1141--1150 (2012; Zbl 1262.18005), Example 8.13.5], which completes the classification of odd-dimensional MTCs up to rank 23. It is also shown that an odd-dimensional MTC of rank 25 is either pointed, perfect or equivalent to the representation category \(\mathrm{Rep}\left( D^{\omega}\left( \mathbb{Z}_{7}\rtimes \mathbb{Z}_{3}\right) \right) \). For higher ranks, partial results for classification of odd-dimensional MTCs of rank at most 73 are obtained.