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)

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].

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.

[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


2 Answers 2


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.

Rowell, Eric; Stong, Richard; Wang, Zhenghan, On classification of modular tensor categories, Commun. Math. Phys. 292, No. 2, 343-389 (2009). ZBL1186.18005.

  • 1
    $\begingroup$ I made a mistake in my post (I confused l and l+2), now it is fixed. You answer Question 1. Next, Question 2 is about a zero entry in each column except one, so the motivation is intact, and the existence of an example is open to me. $\endgroup$ May 5 at 20:30

Here is an anwser to Question 2 given by Andrew Schopieray by email:

<< Having a zero in your column (or row) is preserved under Galois conjugacy of characters. All columns having a zero but the unit implies the categorical/Frobenius dimension character is fixed under Galois conjugation, hence such an example must be integral. So this is equivalent to asking whether there is a simple integral modular tensor category which is not pointed. >>


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