**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