Nondegenerate extensions of near-group
braided fusion categories

The basic object in what follows is a fusion category [5, Section 2]. Our exposition roughly follows the notation and language used in the standard textbook [4] which we will periodically cite. Fusion categories (over $\mathbb{C}$) are $\mathbb{C}$-linear, semisimple, rigid monoidal categories (with product $\otimes$, unit $\mathbbm{1}$, and duality ^∗), which have finitely many isomorphism classes of simple objects and simple monoidal unit. The canonical examples of fusion categories are $\mathrm{Vec}_{G}$, the category of $G$-graded complex vector spaces, and $\mathrm{Rep}(G)$, the category of finite-dimensional complex representations of $G$, for any finite group $G$. The concepts in the remainder of this paragraph are based on the underlying Grothendieck ring of a fusion category [4, Chapter 3]. We denote the set of isomorphism classes of simple objects of a fusion category $\mathcal{C}$ by $\mathcal{O}(\mathcal{C})$. The decomposition of $x\otimes y$ into simple objects for any $x,y\in\mathcal{O}(\mathcal{C})$ are referred to as the fusion rules of $\mathcal{C}$, and are encoded in the fusion matrices $N_{x}:=(N_{x,y}^{z})_{y,z\in\mathcal{O}(\mathcal{C})}$ where $N_{x,y}^{z}:=\dim_{\mathbb{C}}\mathrm{Hom}_{\mathcal{C}}(x\otimes y,z)$. The largest real eigenvalue of $N_{x}$ is known as the Frobenius-Perron dimension of $x$, or $\mathrm{FPdim}(x)$ for brevity, while the sum of $\mathrm{FPdim}(x)^{2}$ over all $x\in\mathcal{O}(\mathcal{C})$ will be denoted $\mathrm{FPdim}(\mathcal{C})$. A simple object $x\in\mathcal{O}(\mathcal{C})$ is called invertible if $\mathrm{FPdim}(x)=1$ which implies $x\otimes y\in\mathcal{O}(\mathcal{C})$ for all $y\in\mathcal{O}(\mathcal{C})$. When all $x\in\mathcal{O}(\mathcal{C})$ are invertible, we say $\mathcal{C}$ is pointed, and when $\mathrm{FPdim}(x)\in\mathbb{Z}$ for all simple $x$ in $\mathcal{C}$, we say $\mathcal{C}$ is integral. But in general, $\mathcal{C}_{\mathrm{pt}}$ and $\mathcal{C}_{\mathbb{Q}}$ will be the maximal pointed and integral fusion subcategories of $\mathcal{C}$, respectively.

The fusion categories $\mathrm{Vec}_{G}$ for finite groups $G$ are both integral and pointed, while the fusion categories $\mathrm{Rep}(G)$ are just integral unless $G$ is abelian. Both of these families of examples have commutative fusion rules. Moreso, for all objects $x,y$, one can choose natural isomorphisms $c_{x,y}:x\otimes y\to y\otimes x$ satisfying braid-like compatibilities; a fusion category along with a choice of natural isomorphisms $\{c_{x,y}\}_{x,y\in\mathcal{C}}$ satisfying the conditions in [4, Definition 8.1.1] is known as a braided fusion category. If $\{c_{x,y}\}_{x,y\in\mathcal{C}}$ is a braiding on a fusion category $\mathcal{C}$, then $\{c_{y,x}^{-1}\}_{x,y\in\mathcal{C}}$ is also a braiding on $\mathcal{C}$; we denote this braided fusion category by $\mathcal{C}^{\mathrm{rev}}$.

If $\rho:G\to\mathrm{GL}(V)$ and $\sigma:G\to\mathrm{GL}(W)$ are finite-dimensional complex representations of a finite group $G$, then $\rho\otimes\sigma$ and $\sigma\otimes\rho$ are naturally isomorphic by simple transposition of bases of $V\otimes W$ and $W\otimes V$. If we denote each transposition by $c_{\rho,\sigma}$, then $\mathrm{Rep}(G)$ along with $\{c_{\rho,\sigma}\}_{\rho,\sigma\in\mathrm{Rep}(G)}$ is a braided fusion category. In this case $c_{\sigma,\rho}c_{\rho,\sigma}=\mathrm{id}_{\rho,\sigma}$ for all $\rho,\sigma\in\mathrm{Rep}(G)$ and we say that $\mathrm{Rep}(G)$ with this particular braiding is symmetrically braided [4, Section 9.9]. But there are potentially many other symmetric braidings one can equip the fusion category $\mathrm{Rep}(G)$ with. In particular, if $z\in Z(G)$ has order 2, then one can twist the standard symmetric braiding by the parity of the action of $z$ to produce another symmetric braiding [4, Example 9.9.1]. To differentiate between these braided fusion categories, we will denote them by $\mathrm{Rep}(G,z)$, and $\mathrm{Rep}(G)$ will be reserved solely for the underlying fusion category. For uniformity, $\mathrm{Rep}(G)$ equipped with the trivial symmetric braiding will be denoted $\mathrm{Rep}(G,e)$ and we refer to any braided fusion category braided equivalent to $\mathrm{Rep}(G,e)$ as Tannakian.

Any fusion subcategory $\mathcal{D}\subset\mathcal{C}$ of a braided fusion category $\mathcal{C}$ is a braided fusion category with the braiding restricted to $\mathcal{D}$ from $\mathcal{C}$. We define the relative centralizer of $\mathcal{D}$ in $\mathcal{C}$, denoted $C_{\mathcal{C}}(\mathcal{D})$, as the full subcategory of $x\in\mathcal{C}$ such that $c_{y,x}c_{x,y}=\mathrm{id}_{x\otimes y}$ for all $y\in\mathcal{C}$. The special case of $C_{\mathcal{C}}(\mathcal{C})$ is known as the symmetric center of $\mathcal{C}$ as it is clearly a symmetrically braided fusion subcategory of $\mathcal{C}$. In much of the existing literature, the symmetric center of $\mathcal{C}$ is simply denoted $\mathcal{C}^{\prime}$ but we will avoid this notation in the remainder of the exposition as the apostrophe is an overburdened symbol in mathematics. The condition of a braiding being symmetric can be restated as $\mathcal{O}(C_{\mathcal{C}}(\mathcal{C}))=\mathcal{O}(\mathcal{C})$, while we refer to braided fusion categories such that $\mathcal{O}(C_{\mathcal{C}}(\mathcal{C}))=\{\mathbbm{1}\}$ as nondegenerately braided. Of most importance in what follows is that if $\mathcal{D}$ is a braided fusion category and $\mathcal{C}\subset\mathcal{D}$ a fusion subcategory, then [4, Theorem 8.21.5]

For a fixed braided fusion category, often one wants to consider categories containing the original which are larger but have more convenient properties. In this laissez-faire approach, the larger category will be called a cover because the larger category has been loosely thrown on top without any specific information about how the smaller category is being contained. When control is needed, the data of a cover along with specific instructions on how to connect the category to its cover will be called an extension.

Adjectives which apply to braided fusion categories apply to covers and extensions in a natural way. For example, a nondegenerate cover of $\mathcal{C}$ would be a nondegenerately braided fusion category $\mathcal{D}$ which is also a cover of $\mathcal{C}$.

Our description of extensions in Definition 2.2.2 is essentially taken from [14, Definition 4.1] which emphasizes the importance of the embedding. The original definition [16, Conjecture 5.2] in the setting of modular tensor categories aligns with our definition of a cover. In a majority of the literature, there is a strong emphasis on studying nondegenerate covers or extensions $\mathcal{C}\subset\mathcal{D}$ which have the smallest possible Frobenius-Perron dimension, which by Equation (1) is $\mathrm{FPdim}(\mathcal{D})=\mathrm{FPdim}(\mathcal{C})\mathrm{FPdim}(C_{\mathcal{C}}(\mathcal{C}))$. This definition is equivalent to ensuring that no simple object is added in the cover which centralizes the original category. In this way, such covers can be thought of as the optimal vehicle for applying the results of nondegenerately braided fusion categories to arbitrary braided fuson categories.

In the literature thus far, extensions satisfying the hypotheses of Lemma 2.2.3 have been referred to as minimal nondegenerate extensions or minimal modular extensions, with the addition of spherical structures (see Section 2.3). It has been shown that these extensions have additional algebraic structure in the unitary setting [14, Theorem 1.1] as well as physical meaning in the study of topological phases of matter [14, Section 2]. But for many braided fusion cateogries the entire study of minimal nondegenerate extensions in this sense is vaccuous, as none exist. We will retain this entrenched language for both covers and extensions with a hope that the notion of minimal extensions will eventually apply to all braided fusion categories.

There are two structural tools for fusion and braided fusion categories that will be used frequently.

Firstly [4, Section 4.14], each fusion category $\mathcal{C}$ (moreover, every fusion ring) possesses a universal grading, i.e. an additive decomposition of $\mathcal{C}$ by graded components corresponding to elements of a finite group $G$, such that the fusion rules of $\mathcal{C}$ respect the operation of $G$. This grading is faithful in the sense that no graded component is empty, and we will refer to $G$ as the universal grading group of $\mathcal{C}$. It follows from the definition of the universal grading that the trivial component, the fusion subcategory of $\mathcal{C}$ $\otimes$-generated by $x\otimes x^{\ast}$ for all $x\in\mathcal{O}(\mathcal{C})$, is a fusion subcategory which we denote by $\mathcal{C}_{\mathrm{ad}}$ and refer to as the adjoint subcategory [4, Section 4.14]. In this way, if the universal grading group of $\mathcal{C}$ is a finite group $G$, we say $\mathcal{C}$ is a $G$-graded extension of $\mathcal{C}_{\mathrm{ad}}$. If a fusion category $\mathcal{C}$ is faithfully graded by a finite group $G$, denote the graded components $\mathcal{C}_{g}$ for $g\in G$. We will repeatedly use the fact that all graded components are the same dimension [4, Theorem 3.5.2]. In particular, $\mathrm{FPdim}(\mathcal{C}_{\mathrm{ad}})=\sum_{x\in\mathcal{O}(\mathcal{C})\cap\mathcal{C}_{g}}\mathrm{FPdim}(x)^{2}$ for all $g\in G$ and moreover $\mathrm{FPdim}(\mathcal{C})=|G|\mathrm{FPdim}(\mathcal{C}_{\mathrm{ad}})$.

Secondly, if $\mathcal{C}$ is a fusion category and $\mathrm{FPdim}(\mathcal{C})\in\mathbb{Z}$ (i.e. $\mathcal{C}$ is weakly integral), then $\mathcal{C}$ possesses a canonical positive spherical structure [4, Corollary 9.6.6]. In practice, we only need this structure to allow us to take traces of endomorphisms in $\mathcal{C}$, producing numerical constraints in our proofs. The positivity of the canonical spherical structure corresponds to the fact that $\dim(x)$, the traces of the identity morphisms on $x\in\mathcal{O}(\mathcal{C})$, are precisely $\mathrm{FPdim}(x)$. This implies $\mathrm{FPdim}(\mathcal{C})=\dim(\mathcal{C}):=\sum_{x\in\mathcal{O}(\mathcal{C})}\dim(x)^{2}$ hence in the setting of weakly integral fusion categories we will use $\mathrm{FPdim}$ and $\dim$ interchangeably, while for arbitrary fusion categories we will only use $\mathrm{FPdim}$. We will consider nondegenerately braided fusion categories as modular tensor categories [4, Definition 8.13.4] equipped with their unique positive spherical structure, when it suits our purposes, in order to utilize their modular data [4, Section 8.17]. Conveniently, the universal grading group of a modular tensor category $\mathcal{C}$ is canonically isomorphic to $\mathcal{O}(\mathcal{C}_{\mathrm{pt}})$ [11, Theorem 6.3]. The modular data of a modular tensor category is an invertible $|\mathcal{O}(\mathcal{C})|\times|\mathcal{O}(\mathcal{C})|$ matrix $S=(S_{x,y})_{x,y\in\mathcal{O}(\mathcal{C})}$ consisting of the traces of the double-braidings [4, Section 8.13], and a diagonal matrix with diagonal elements $(\theta_{x})_{x\in\mathcal{O}(\mathcal{C})}$ where $\theta_{x}$ is the trace of the ribbon structure $\mathcal{C}$. The matrix $S$ is unitary [5, Proposition 2.12]; we will refer to this fact as orthogonality relations between simple objects of $\mathcal{C}$. Specifically, if $x,y\in\mathcal{O}(\mathcal{C})$, then $\sum_{z\in\mathcal{O}(\mathcal{C})}S_{x,z}\overline{S_{y,z}}=\dim(\mathcal{C})$ if $x\cong y$, and is zero otherwise, using $\overline{\alpha}$ as the complex conjugate of $\alpha\in\mathbb{C}$. It is known that $\theta_{x}$ are roots of unity for all $x\in\mathcal{O}(\mathcal{C})$ [4, Corollary 8.18.2] and we will refer to the order of the $T$-matrix as the conductor of $\mathcal{C}$. Two formulas that will be used in our proofs are the Verlinde formula [4, Corollary 8.14.4], which states that the fusion rules of a modular tensor category are given by the $S$-matrix via

for all $w,y,z\in\mathcal{O}(\mathcal{C})$, and the balancing equation [4, Proposition 8.13.8], which states that even without nondegeneracy of the braiding,

The last benefit of the $S$-matrix is that it allows a numerical test for two objects $x,y$ in a spherical braided fusion category $\mathcal{C}$ to centralize one another, i.e. $c_{y,x}c_{x,y}=\mathrm{id}_{x\otimes y}$. By taking the trace of these endomorphisms we can see $x,y$ centralizing one another implies $S_{x,y}=\dim(x)\dim(y)$, and the converse is also true [16, Proposition 2.5].

Let $p$ be a prime integer. Recall that a finite group $G$ is a $p$-group if the order of $G$ is a power of $p$. A $p$-group $G$ is extraspecial if $|Z(G)|=p$ and $G/Z(G)$ is a non-trivial elementary abelian $p$-group, i.e. isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{\oplus n}$ for some $n\in\mathbb{Z}_{\geq 1}$. For each prime integer $p$ and positive integer $n$, there exist exactly two isomorphism classes of extraspecial $p$-groups of order $p^{2n+1}$ which are traditionally denoted $p^{1+2n}_{\pm}$. There do not exist extraspecial $p$-groups whose order is an even power of $p$. All extraspecial $p$-groups can be constructed as central products of the two extraspecial $p$-groups of order $p^{3}$. For example, the extraspecial $2$-groups of order 8 are the dihedral group $D_{4}=2^{1+2}_{+}$ and the quaternion group $Q_{8}=2^{1+2}_{-}$. Therefore, all extraspecial $2$-groups are just central products of various copies of $D_{4}$ and $Q_{8}$.

The representation theory of extraspecial $p$-groups is straightforward as $p^{2n}$ isomorphism classes of irreducible representations of $p^{1+2n}_{\pm}$ are one-dimensional, corresponding to the elementary abelian $p$-group underlying $p^{1+2n}_{\pm}$. We will abuse notation and denote these classes by $g$ for $g\in(\mathbb{Z}/p\mathbb{Z})^{\oplus 2n}$ whose fusion rules are simply those of the linear characters of $(\mathbb{Z}/p\mathbb{Z})^{\oplus 2n}$. The remaining $p-1$ isomorphism classes of irreducible representations are faithful of dimension $p^{n}$, and are distinguished by their values on $Z(p^{1+2n}_{\pm})$. Denote these classes by $x_{g}$ for $g\in(\mathbb{Z}/p\mathbb{Z})^{\times}$. The fusion rules involving $x_{h}$ are then

Extraspecial $p$-groups are characterized by the degrees of their characters. This result can be found in standard textbooks on the character theory of finite groups such as [13, Proposition 7.7]. For our purposes, we will only need a trivial corollary of this fact.

Tambara-Yamagami fusion categories are $\mathbbm{Z}/2\mathbb{Z}$-graded extensions (see Section 2.3) of pointed fusion categories whose non-trivial graded component has exactly one isomorphism class of simple objects. We will denote the isomorphism classes of invertible objects by $g\in G$, a finite group, and the isomorphism class of the noninvertible simple object by $x$. The fusion rules for invertible elements follow the group operation of $G$ while fusion rules with $x$ must be $g\otimes x\cong x\otimes g\cong x$ and $x\otimes x\cong\bigoplus_{g\in G}g$. For example, $\mathrm{Rep}(2_{\pm}^{1+2n})$ for $n\in\mathbb{Z}_{\geq 1}$ are Tambara-Yamagami fusion categories whose adjoint subcategories have the fusion rules of an elementary abelian $2$-group. It was shown in [23] that a Tambara-Yamagami fusion category (over $\mathbb{C}$) is characterized by the group of invertible objects, $G$, which must be abelian, a nondegenerate bicharacter $\chi:G\times G\to\mathbb{C}^{\times}$ such that $\chi(g,h)=\chi(h,g)$ for all $g,h\in G$, and an extension sign [6, Example 9.4] which is usually recorded as $\tau$, a square root of $|G|^{-1}$ as this value appears in many computations. It is reasonable to use the notation $\mathcal{C}(\chi,\tau)$ for such a category since a description of $\chi$ includes a description of the finite abelian group $G$. This is the original notation of D. Tambara and S. Yamagami [23, Definition 3.1].

It has been proven [21, Theorem 1.2(1)] that Tambara-Yamagami fusion categories possess a braiding if and only if $G$ is an elementary abelian 2-group, i.e. $G=E_{n}:=(\mathbb{Z}/2\mathbb{Z})^{\oplus n}$ for some $n\in\mathbb{Z}_{\geq 1}$. Up to isometry, there is a unique nondegenerate bicharacter $\chi^{1}$ on $E_{n}$ such that $\chi^{1}(g,h)=\chi^{1}(h,g)$ for all $g\in E_{n}$ when $n$ is odd and exactly two, $\chi^{0},\chi^{1}$, up to isomorphism when $n$ is even [25, Section 5]. Specifically, $\chi^{0}(g,g)=1$ and $\chi^{1}(g,g)=-1$ for all generators $g\in E_{n}$. When no specific braiding needs to be defined, it will suffice to denote all braided Tambara-Yamagami fusion categories as $\mathcal{C}(\chi^{k}_{n},\tau)$ where $k=1,2$ and $n$ indicates $E_{n}$ is the underlying group of invertible objects.

For fixed $n\in\mathbb{Z}_{\geq 1}$, it was further proven that there exist at most $2^{n+1}$ inequivalent braidings for each $\mathcal{C}(\chi_{n}^{k},\tau)$, indexed by choices of signs $(\delta_{1},\ldots,\delta_{n},\epsilon)$ [21, Theorem 1.2(2)]. Many of these braidings are equivalent. A more precise statement can be found in [7, Corollary 4.10] and elsewhere. Here the data $(\delta_{1},\ldots,\delta_{n},\epsilon)$ is replaced with $(q,\alpha)$ where $q:E_{n}\to\mathbb{C}^{\times}$ is a quadratic form such that $\chi_{n}^{k}(g,h)=q(g)q(h)q(g+h)^{-1}$ for all $g,h\in E_{n}$, and $\alpha\in\mathbb{C}$ is a chosen square root of $\tau\sum_{g\in E_{n}}q(g)$. To translate between the two sets of data, $q(g)=\sigma_{1}(g)$ for all $g\in E_{n}$ in the notation of [21, Section 2.3], while $\alpha=\sigma_{3}(e)$. Two braidings $(q,\alpha)$, $(q^{\prime},\alpha^{\prime})$ on a fixed fusion category $\mathcal{C}(\chi,\tau)$ with $\chi:E_{n}\times E_{n}\to\mathbb{C}^{\times}$ are equivalent if and only if there exists a group automorphism $f\in\mathrm{Aut}(E_{n})$ such that $q^{\prime}(f(g))=q(g)$ for all $g\in E_{n}$ and $\alpha=\alpha^{\prime}$ [7, Corollary 4.10]. This implies that $f$ satisfies $\chi(g,h)=\chi(f(g),f(h))$ for all $g,h\in E_{n}$.

Lastly, we outline the center, or double construction, for fusion categories and its relation to commutative algebras. If $\mathcal{C}$ is a fusion category, then $\mathcal{Z}(\mathcal{C})$, the center or double of $\mathcal{C}$ has objects in correspondence with pairs $(x,\{\rho_{y}\}_{y\in\mathcal{C}})$ where $x\in\mathcal{C}$ and $\rho_{y}:x\otimes y\to y\otimes x$ is a natural isomorphism for all $y\in\mathcal{C}$ satisfying the coherence diagram in [4, Definition 7.13.1] where one can find the definition of morphisms in this category, its monoidal product, etc. The goal of this construction is to create a nondegenerately braided fusion category from $\mathcal{C}$ in a uniform manner, which often ends up being quite unwieldly. The namesake of the double comes from the fact that the representation category of the quantum double of a Hopf algebra $H$ is precisely the double $\mathcal{Z}(\mathrm{Rep}(H))$ [4, Proposition 7.14.6]. Let $G$ be a finite group and $\omega\in H^{3}(G,\mathbb{C}^{\times})$ a cohomological twisting of the associativity of $\mathrm{Vec}_{G}$, producing the fusion category of twisted $G$-graded vector spaces $\mathrm{Vec}_{G}^{\omega}$ [4, Example 2.3.8]. The center $\mathcal{Z}(\mathrm{Vec}_{G}^{\omega})$ is known as the twisted double of $G$ whose modular data has been known for quite some time [9, Sections 2.2 & 5.2].

Alternatively, twisted doubles of finite groups can be characterized by possessing maximal Tannakian fusion subcategories which we will understand through the following construction. In general, if $\mathcal{C}$ is a braided fusion category and $\mathrm{Rep}(G)\subset\mathcal{C}$ is a Tannakian fusion subcategory for a finite group $G$, then $R:=\oplus_{g\in G}g$ has the structure of a connected étale algebra in $\mathcal{C}$ [1, Section 3]; we will refer to this algebra as the regular algebra of the Tannakian subcategory. The category of local $R$-modules $\mathcal{C}_{R}^{0}$ is then a braided fusion category which is nondegenerately braided if and only if $\mathcal{C}$ is [1, Corollary 3.30]. Moreover $\mathcal{C}_{R}^{0}$ inherits the spherical structure of $\mathcal{C}$ so the passage $\mathcal{C}\to\mathcal{C}_{R}^{0}$ sends modular tensor categories to modular tensor categories. When $R$ is the regular algebra of a Tannakian fusion subcategory $\mathcal{E}\subset\mathcal{C}$, then the simple objects of $\mathcal{C}_{R}^{0}$ are summands of the free $R$-modules $R\otimes x$ for $x\in\mathcal{O}(C_{\mathcal{C}}(\mathcal{E}))$ [1, Example 3.14].

We will use regular algebras of Tannakian subcategories for numerical arguments as well, but one important application is to prove a reconstruction theorem for twisted doubles of finite groups from Lagrangian subcategories, i.e. Tannakian fusion subcategories $\mathcal{E}$ of braided fusion categories $\mathcal{C}$ such that $\mathrm{FPdim}(\mathcal{E})^{2}=\mathrm{FPdim}(\mathcal{C})$. Theorem 4.5 of [2] states that a modular tensor category $\mathcal{C}$ is braided equivalent to a twisted double of a finite group $G$ if and only if there exists a Lagrangian subcategory $\mathrm{Rep}(G)\subset\mathcal{C}$.

Let $\mathcal{C}$ be one of the 4 nonsymmetrically braided fusion categories $\mathcal{C}(\chi^{0}_{2},\tau)$ (Figure 4). Note that the symmetric center $C_{\mathcal{C}}(\mathcal{C})=\mathcal{C}_{\mathrm{pt}}$ is Tannakian, using the formulas of [21, Section 2.3(2)]. Let $\mathcal{D}$ be a nondegenerate cover of $\mathcal{C}$ with $\mathrm{FPdim}(\mathcal{D})=32$. As $\mathcal{D}$ is weakly integral, we may assume $\mathcal{D}$ is a modular tensor category equipped with its unique positive spherical structure. We will refer to $\mathcal{O}(\mathcal{C}_{\mathrm{pt}})$ by its structure as an abelian group, $E_{2}=(\mathbb{Z}/2\mathbb{Z})^{2}$.

We will first demonstrate that $\mathcal{D}_{\mathrm{pt}}=\mathcal{C}_{\mathrm{pt}}$ and $\mathcal{D}_{\mathrm{ad}}=\mathcal{C}$. To this end, since $\mathcal{D}$ is weakly integral, then [10, Lemma 1.1] implies $\mathcal{D}_{\mathrm{ad}}$ is integral, of dimension $4$ or $8$ since $\dim(\mathcal{D})=\dim(\mathcal{D}_{\mathrm{pt}})\dim(\mathcal{D}_{\mathrm{ad}})$ by [4, Corollaries 8.21.7 & 8.22.8]. Recall that $\mathcal{D}$ is nilpotent, as its dimension is a prime power [11, Example 4.5]. Hence in the former case, every $y\in\mathcal{O}(\mathcal{D})$ satisfies $\dim(y)^{2}\in\{1,2,4\}$ as $\dim(y)^{2}$ must divide $\dim(\mathcal{D}_{\mathrm{ad}})$ [11, Theorem 5.2.]. If there exists simple $y\in\mathcal{D}$ with $\dim(y)^{2}=2$ then there exists a unique simple object $g\in E_{2}$ with $g\otimes y\cong y$: the nontrivial summand of $y\otimes y^{\ast}$. The other two nontrivial $h_{1},h_{2}\in E_{2}$ must then satisfy $h_{j}\otimes y\cong z$ where $z$ is another simple object in the same universally graded component as $y$ with $\dim(z)^{2}=2$. Morever $y\otimes z\cong h_{1}\oplus h_{2}\in\mathcal{D}_{\mathrm{ad}}$. Therefore $y\otimes y\cong\mathbbm{1}\oplus g$ (and $z\otimes z\cong\mathbbm{1}\oplus g$), implying $y\cong y^{\ast}$. This forces $y$ to $\otimes$-generate a braided fusion category of dimension 4 which is not pointed. Such a category is an Ising modular tensor category, whose pointed subcategory is not Tannakian [3, Lemma B.18]. But $\mathcal{C}_{\mathrm{pt}}$ is Tannakian so this is cannot be the case. Hence every simple object is invertible or dimension 2 and thus lies in its own universally graded component. This implies $g\otimes x\cong x$ for any $x$ such that $\dim(x)=2$, hence $S_{g,x}=2$ by the balancing equation and thus $x$ is centralized by $\mathcal{D}_{\mathrm{ad}}$. This is a contradiction since $C_{\mathcal{D}}(\mathcal{D}_{\mathrm{ad}})=\mathcal{D}_{\mathrm{pt}}$. We may then conclude that $\dim(\mathcal{D}_{\mathrm{ad}})=8$, hence $\mathcal{D}_{\mathrm{pt}}=\mathcal{C}_{\mathrm{pt}}$, and lastly $\mathcal{D}_{\mathrm{ad}}=\mathcal{C}$.

We may now reveal the $S$-matrix of $\mathcal{D}$ (Section 2.3). The balancing equation implies $S_{x,x}=S_{x,x^{\ast}}=4\theta_{x}^{-2}$ since $\theta_{g}=1$ for all $g\in E_{2}$, while $S_{x,g}=2$ for all $g\in E_{2}$ since $x$ centralizes $g$. From the formulas of [21, Section 3.7] and [21, Section 2.3(2)], $\theta_{x}=\pm i$ since $\mathcal{C}$ is not symmetric. Moreover $S_{x,x}=-4$. The orthogonality relation (Section 2.3) of $x$ with itself is

Therefore $\sum_{z\in\mathcal{O}(\mathcal{D})\setminus\mathcal{O}(\mathcal{C})}|S_{x,z}|^{2}=0$ and moreover $S_{x,z}=0$ for all $z\in\mathcal{O}(\mathcal{D})\setminus\mathcal{O}(\mathcal{C})$. Verlinde formula then implies for $y\in\mathcal{O}(\mathcal{D})\setminus\mathcal{O}(\mathcal{C})$,

since for every $z\in\mathcal{O}(\mathcal{D})\setminus E_{2}$, we have shown either $S_{x,z}=0$ or $S_{y,z}=0$. Moreover, we have computed the remaning $S$-matrix entries above, implying