# Simple modular tensor category and zero entries in its S-matrix

Question 1: Is there a simple modular fusion category with a zero entry in its S-matrix?
(or equivalently, with a fusion matrix of zero determinant?)

Yes, by this answer below providing the example $$\mathrm{PSU}(2)_{7}$$. Moreover, the checking paragraph below suggests that $$\mathrm{PSU}(2)_{\ell}$$ is such an example if and only if $$\ell+2$$ is odd and composed.

Question 2: Is there a simple modular fusion category with a zero entry in each column of its S-matrix, except one?
(or equivalently, with every fusion matrix of zero determinant, except one)

Checking
In the pointed case, the fusion matrices are permutation matrices, so of determinant $$\pm 1$$. The easiest non-pointed simple modular fusion categories are $$\mathrm{PSU}(2)_{\ell}$$, with $$\ell$$ odd. It is a fusion subcategory of $$\mathrm{SU}(2)_{\ell}$$, and [BK01, Example 3.3.22] implies that the S-matrix of $$\mathrm{SU}(2)_{\ell}$$ has no zero entry iff $$\ell + 2$$ is prime. In particular, the S-matrix of $$\mathrm{PSU}(2)_{\ell}$$ has no zero entry if $$\ell + 2$$ is prime. Now by [BK01, Example 3.3.22] again, for all $$\ell$$, the S-matrix of $$\mathrm{SU}(2)_{\ell}$$ does not have a zero entry in each column (except one), thus so is for $$\mathrm{PSU}(2)_{\ell}$$, and we are done. The S-matrix for a modular fusion category of Lie type in general is given by [BK01, Theorem 3.3.20], and examples may be found there. The simple modular fusion category from [Sch22] has also all its fusion matrices of determinant $$\pm 1$$. A good example to check would be the center of Extended Haagerup [MW17].

Motivation
A negative answer to Question 2 can be formulated as follows:
Statement 1 (Open): Every simple modular fusion category has no zero entry in each column (except one) of its S-matrix.
(or equivalently, has no every fusion matrix of zero determinant, except one).

Observe that Statement 1 and [GNN09, Theorem 6.1] (or, more generally, [Bur23, Theorem 2]) imply:
Statement 2 (Open): Every simple integral modular fusion category is pointed.

Next, Statement 2 and [LPR23, Theorem 5.8] imply:
Statement 3 (Open): Every simple integral fusion category is weakly group-theoretical.

Conclusion: a negative answer to above question provides a negative answer to [ENO11, Question 2] in the simple case.

References
[BK01] B. Bakalov, Bojko A. Kirillov, Lectures on tensor categories and modular functors. University Lecture Series, 21. AMS, 2001.
[Bur23] S. Burciu, On the Galois symmetries for the character table of an integral fusion category. J. Algebra Appl. 22 (2023), no. 1, Paper No. 2350026.
[ENO11] P. Etingof, D. Nikshych, and V. Ostrik, Weakly group-theoretical and solvable fusion categories, Adv. Math., 226 (2011), pp. 176–205.
[GNN09] S. Gelaki, D. Naidu, D. Nikshych, Centers of graded fusion categories. Algebra Number Theory 3 (2009), no. 8, 959--990.
[LPR23] Z. Liu, S. Palcoux, Y. Ren, Interpolated family of non group-like simple integral fusion rings of Lie type, Internat. J. Math. (2023), DOI: 10.1142/S0129167X23500301.
[MW17] S. Morrison, K. Walker, The center of the extended Haagerup subfactor has 22 simple objects. Internat. J. Math. 28 (2017), no. 1, 1750009.
[Sch22] A. Schopieray, Non-pseudounitary fusion. J. Pure Appl. Algebra 226 (2022), no. 5, Paper No. 106927

• For the symmetric case (more precisely, finite group rep) instead of modular, see mathoverflow.net/q/401924/34538 May 5 at 20:36

I may be misunderstanding the definition of simple (I am using your post Strongly simple fusion categories: the known examples? as reference), but I believe that the $$(A_1,7)_{1/2}$$ quantum group MTC is an example (see Wang et. al's classification paper cited below, formula 5.3.10). The $$S$$-matrix has a zero entry on its diagonal.
It is simple because the fusion rule for $$\alpha^2$$ involves $$\omega$$, the fusion rule for $$\omega^2$$ involves $$\rho$$, and the fusion rule for $$\rho^2$$ involves $$\alpha$$. Hence, you can't pick a subset of the simple objects and have the fusion be self contained. Hence, $$(A_1,7)_{1/2}$$ should have no non-trivial fusion subcategories so it should be simple.