2 Preliminaries

In this paper, we work over the base field $\mathbb {C}$ . For a comprehensive overview of fusion categories, we direct the reader to the monograph [Reference Etingof, Gelaki, Nikshych and Ostrik13].

Let $\mathcal {C}$ be a fusion category. We denote the maximal pointed fusion subcategory of $\mathcal {C}$ by ${\mathcal {C}_{\mathrm {pt}}}$ , and let $\mathrm {Irr}(\mathcal {C})$ be a set of representatives for the isomorphism classes of simple objects in $\mathcal {C}$ . The size of $\mathrm {Irr}(\mathcal {C})$ is referred to as the rank of $\mathcal {C}$ and is denoted by $\operatorname {\mathrm {{rank}}}(\mathcal {C})$ .

A pivotal structure in a rigid monoidal category $\mathcal {C}$ is an isomorphism $j:\operatorname {\mathrm {id}}_{\mathcal {C}}\to ()^{**}$ between monoidal functors. A pivotal monoidal category refers to a rigid monoidal category equipped with such a pivotal structure. Recall from [Reference Müger20] that a pivotal fusion category is spherical if and only if $\dim (V ) = \dim (V^*)$ for all simple objects V, where $\dim $ denotes the quantum dimension of $\mathcal {C}$ . A fusion category $\mathcal {C}$ is called pseudo-unitary if its Frobenius–Perron dimension $\mathrm {FPdim}(\mathcal {C})$ equals its categorical dimension $\dim (\mathcal {C})$ . In this case, according to [Reference Etingof, Nikshych and Ostrik14, Proposition 8.23], the category $\mathcal {C}$ possesses a unique spherical structure, known as the canonical spherical structure, under which the categorical dimensions of all simple objects are positive. With this spherical structure, the categorical dimension of any object X in $\mathcal {C}$ matches its Frobenius–Perron dimension (i.e., $\mathrm {FPdim}(X) = \dim (X)$ for every object X).

Recall that a fusion category is called weakly integral if its Frobenius-Perron dimension is an integer. Moreover, the fusion category is called integral if the Frobenius-Perron dimension of each object is an integer. Any weakly integral fusion category is inherently pseudo-unitary and possesses a canonical spherical structure, under which the categorical dimensions match the Frobenius–Perron dimensions [Reference Etingof, Nikshych and Ostrik14, Proposition 8.24].

For any fusion category $\mathcal {C}$ , the monoidal center (or the Drinfeld center) of $\mathcal {C}$ is a braided fusion category ${\mathcal {Z}(\mathcal {C})}$ equipped with a forgetful functor $F:{\mathcal {Z}(\mathcal {C})}\rightarrow \mathcal {C}$ ; see [Reference Etingof, Gelaki, Nikshych and Ostrik13].

The forgetful functor F also admits a right adjoint functor $R:\mathcal {C} \rightarrow {\mathcal {Z}(\mathcal {C})}$ , and $Z :=FR:\mathcal {C} \rightarrow \mathcal {C}$ is a Hopf comonad defined as an end. Indeed, following [Reference Shimizu22, Section 2.6], one has that

(2.1) $$ \begin{align} Z(V)\simeq \int_{X\in \mathcal{C}}X\otimes V\otimes X^*. \end{align} $$

It is known that $A:=Z(\mathbf {1})$ (the image of the unit object) has the structure of a (central) commutative algebra in $\mathcal {Z}(\mathcal {C})$ , and it is called the adjoint algebra of $\mathcal {C}$ .

The vector space $\mathrm { {CE}(\mathcal {C})}:= \mathrm {{Hom}}_{\mathcal {C}}(\mathbf {1}, A) $ is referred to as the the set of central elements. The vector space ${\mathrm {CF}(\mathcal {C})}:=\mathrm {{Hom}}_{\mathcal {C}}(A,\mathbf {1})$ is referred to as the space of class functions of $\mathcal {C}$ .

Note that for a fusion category over $\mathbb {C}$ , by [Reference Shimizu22, Example 4.4], there is a $\mathbb {C}$ -algebra isomorphism ${\mathrm {CF}(\mathcal {C})}\simeq {K(\mathcal {C})}\otimes _{\mathbb {Z}}\mathbb {C}$ . Therefore, ${\mathrm {CF}(\mathcal {C})}$ is commutative if and only if ${K(\mathcal {C})}$ is commutative.

Both ${\mathrm {CF}(\mathcal {C})}$ and $\mathrm { {CE}(\mathcal {C})}$ have a semisimple $\mathbb {C}$ -algebra structure, as described in [Reference Shimizu22]. Additionally, $\mathrm { {CE}(\mathcal {C})}$ is a commutative algebra, which means it decomposes as a direct product of copies of the field $\mathbb {C}$ .

Since $R:\mathcal {C} \rightarrow {\mathcal {Z}(\mathcal {C})}$ is a right adjoint to the forgetful functor $F:{\mathcal {Z}(\mathcal {C})} \rightarrow \mathcal {C}$ , [Reference Shimizu22, Theorem 3.8] establishes that this adjunction induces an isomorphism of $\mathbb {C}$ -algebras:

(2.2) $$ \begin{align} {\mathrm{CF}(\mathcal{C})} \xrightarrow{\cong} \mathrm{End}_{{\mathcal{Z}(\mathcal{C})}}(R(\mathbf{1})),\;\; \chi\mapsto Z (\chi)\circ \delta_{\mathbf{1}}, \end{align} $$

where $\delta _{\mathbf {1}}$ is the comultiplication of the Hopf comonad Z associated at the unit $\mathbf {1}$ of the fusion category $\mathcal {C}$ .

2.1 The bijection between central primitive idempotents and conjugacy classes

Suppose that $\mathcal {C}$ is a pivotal fusion category with a commutative Grothendieck ring $K(\mathcal {C})$ . Since the monoidal center ${\mathcal {Z}(\mathcal {C})}$ is also a fusion category, we can write $R(\mathbf {1})=\bigoplus _{i=0}^m\mathcal C_i$ as a direct sum of simple objects in ${\mathcal {Z}(\mathcal {C})}$ . Note that any two distinct conjugacy classes are not isomorphic to each other, since from the isomorphism of Equation (2.2), the endomorphism ring of $R(\mathbf {1})$ is a commutative $\mathbb {C}$ -algebra. Thus, $\mathcal C^{0},\dots , \mathcal C^{m}$ represent the conjugacy classes of $\mathcal {C}$ . Since the unit object $\mathbf {1}_{{\mathcal {Z}(\mathcal {C})} }$ is always a sub-object of $R(\mathbf {1})$ , we can assume $\mathcal C^{0} = \mathbf {1}_{{\mathcal {Z}(\mathcal {C})} }$ . In [Reference Shimizu22, Section 5], Shimizu also introduced a canonical pairing between ${\mathrm {CF}(\mathcal {C})}$ and $\mathrm { {CE}(\mathcal {C})}$ with values in $\mathbb {C}$ . This pairing is defined by the condition

(2.3) $$ \begin{align} f\circ a=\langle f, a\rangle\operatorname{\mathrm{id}}_{\mathbf{1}} \end{align} $$

for all $f \in {\mathrm {CF}(\mathcal {C})}$ and $a\in \mathrm { {CE}(\mathcal {C})}$ .

Let $\tilde F_0, \tilde F_1, \dots , \tilde F_m\in \mathrm {End}_{{\mathcal {Z}(\mathcal {C})}}(R(\mathbf {1}))$ be the canonical projections onto each of these conjugacy classes. Let also $F_0, F_1, \dots F_m$ be the corresponding primitive idempotents of ${\mathrm {CF}(\mathcal {C})}$ under the canonical adjunction isomorphism ${\mathrm {CF}(\mathcal {C})} \simeq \mathrm {End}_{{\mathcal {Z}(\mathcal {C})}}(R(\mathbf {1}))$ from Equation (2.2). This establishes a canonical bijection between the (central) primitive idempotents of ${\mathrm {CF}(\mathcal {C})}$ and the conjugacy classes of $\mathcal {C}$ .

Let $\mathcal {J}:=\{0,1,\dots ,m\}$ represent the set of indices corresponding to the central primitive idempotents of ${\mathrm {CF}(\mathcal {C})}$ . To each $F_j$ with $j\in \mathcal J$ , the bijection described above assigns a conjugacy class $\mathcal {C}^j$ .

2.2 On the definition of conjugacy class sums

In a pivotal fusion category, Shimizu defines the elements $C_j := \mathcal {F}_\lambda ^{-1}(F_j) \in \mathrm { {CE}(\mathcal {C})}$ as the conjugacy class sums corresponding to the conjugacy class $\mathcal {C}^j$ . Here, $\lambda \in \mathrm {{CF}}(\mathcal {C})$ is a cointegral of $\mathcal {C}$ such that $\langle \lambda , u \rangle = 1$ (see [Reference Shimizu22, Section 5]). Here, $u:\mathbf {1}\rightarrow A$ is the unit of the central algebra A. Additionally, the Fourier transform of $\mathcal {C}$ associated with $\lambda $ is defined as the linear map

(2.4) $$ \begin{align} \mathcal F_{\lambda}:\mathrm{ {CE}(\mathcal{C})}\rightarrow {\mathrm{CF}(\mathcal{C})}\;\;\text{given by}\;\;a \mapsto \lambda \leftharpoonup \mathcal S(a), \end{align} $$

where $\mathcal {S}$ denotes the antipode.

Example 2.2. If $\mathcal {C} = \mathrm {Vec}^{\omega }_G$ is a pointed fusion category, then each conjugacy class sum of G corresponds to a representation of the group algebra $\mathbb {C}[G]$ . Furthermore, the space of central elements $\mathrm { {CE}(\mathcal {C})}$ is isomorphic to $K(G)$ , the Grothendieck group of $\mathbb {C}[G]$ . In particular, if G is abelian, the conjugacy classes of $\mathcal {C}$ can be identified with elements of G, since in this case, the dual group $\widehat {G}$ is isomorphic to G.

2.3 The integral of a fusion category

Let $\mathcal {C}$ be a fusion category, and let $A=Z(\mathbf {1})$ be its adjoint algebra as defined earlier. Following [Reference Shimizu22, Definition 5.7], an integral in $\mathcal {C}$ is a morphism $\Lambda : \mathbf {1} \rightarrow A$ in $\mathcal {C}$ that satisfies

$$ \begin{align*} m \circ (\operatorname{\mathrm{id}}_{A}\otimes \Lambda)=\epsilon_{\mathbf{1}}\otimes \Lambda, \end{align*} $$

where $m:A\otimes A\rightarrow \mathbf {1}$ is the multiplication of the adjoint algebra A. Here, $\epsilon _{\mathbf {1}}$ is counit $\epsilon _V:Z(V)\rightarrow V $ specialized at $V=\mathbf {1}$ , the unit object of the fusion category $\mathcal {C}$ .

It is known that the integral of a fusion category is unique up to a scalar. Let $\Lambda \in \mathrm { {CE}(\mathcal {C})}$ be the idempotent integral of $\mathcal {C}$ and define $\widehat {\Lambda }:=\dim (\mathcal {C})\Lambda $ .

Let $\{V_0, \dots , V_m\}$ be a complete set of representatives of the isomorphism classes of simple objects in $\mathcal {C}$ . Shimizu, in [Reference Shimizu22, Section 6], provided a construction of the (central) primitive elements $E_i\in \mathrm { {CE}(\mathcal {C})}$ , corresponding to the irreducible characters $\chi _i:=\chi _{V_i}$ associated to each simple object $V_i$ of $\mathcal {C}$ . This correspondence is defined by the condition:

$$ \begin{align*}\langle\chi_i, E_j\rangle=\delta_{i, j},\end{align*} $$

for all $i,j$ .

Suppose now that ${K(\mathcal {C})}$ is commutative. Note that [Reference Burciu3, Lemma 4.1], together with Equation (4.7) from the same paper, provides an alternative pair of dual bases for the canonical pairing. More precisely, it shows that $\{F_j\}$ and $\{\frac {C_j}{\dim (\mathcal {C}^j)}\}$ form another pair of dual bases expressed as

(2.5) $$ \begin{align} \langle F_j, \frac{C_{j'}}{\dim(\mathcal{C}^{{j^{\prime}}})} \rangle=\delta_{j,j'} \end{align} $$

for all $j, j'\in \mathcal J$ .

Without loss of generality, we can assume that $V_0=\mathbf {1}$ , the unit of the fusion category $\mathcal {C}$ . This gives $\chi _0=\epsilon _{\mathbf {1}}$ , and the associated primitive idempotent is $E_0=\Lambda _{\mathcal {C}}$ ; see [Reference Shimizu22, Lemma 6.1].

The following lemma is straightforward, but for reader’s convenience, a proof is provided.

Lemma 2.1. Let $\mathcal {C}$ be a pivotal fusion category with a commutative Grothendieck ring. We have the following formula for the integral element of $\mathcal {C}$ :

(2.6) $$ \begin{align} \widehat{\Lambda}_{\mathcal{C}}=\sum_{j\in \mathcal{J}} C_j. \end{align} $$

Proof. We will show that both sides of the Equation (2.6) have the same image under $\mathcal F_\lambda $ . For the right-hand side, by the definition of conjugacy class sums $C_j$ , we have $\mathcal F_\lambda (\sum _{j\in \mathcal {J}} C_j)=\sum _{j\in \mathcal {J}} F_j=\epsilon _{\mathbf {1}}$ . For the left-hand side, according to [Reference Shimizu22, Equation (6.10)], we have

(2.7) $$ \begin{align} \mathcal F_\lambda(E_{i^*})=\frac{d_i}{\dim(\mathcal{C})}\chi_i, \end{align} $$

where the index $i^*$ is defined such that $V_i^*\simeq V_{i^*}$ . Thus, $E_{i^*}$ is the primitive (central) idempotent corresponding to $\chi _{i^*}=\chi _{V_{i^*}}$ .

In the case $i=0$ , one has $\chi _0=\epsilon _{\mathbf {1}}$ and $E_0=\Lambda _{\mathcal {C}}$ . Since $d_0=1$ , it follows that ${\mathcal F_\lambda (E_{0})=\epsilon _{\mathbf {1}}.}$

2.4 Definition of the support $\mathcal J_{{\mathcal {D}}}$

Let $\mathcal {C}$ be a fusion category whose Grothendieck ring is commutative. For any fusion subcategory ${\mathcal {D}} \subseteq \mathcal {C}$ , recall the notion of its support from [Reference Burciu3, Subsection 4.2]. It is known that there is an inclusion of $\mathbb {C}$ -algebras $\iota : \mathrm {{CF}}({\mathcal {D}}) \hookrightarrow \mathrm {{CF}}(\mathcal {C})$ . Consequently, there exists a subset $\mathcal {J}^{\mathcal {C}}_{\mathcal {D}} \subseteq \mathcal J^{\mathcal {C}}:=\{0, \dots , m\}$ such that

(2.8) $$ \begin{align} \lambda_{{\mathcal{D}}}=\sum_{j \in \mathcal J_{\mathcal{D}}}F_j \end{align} $$

since $\lambda _{\mathcal {D}}$ is an idempotent element in $\mathrm {{CF}}(\mathcal {C})$ . The set $\mathcal {J}_{\mathcal {D}}$ is referred to as the support of the fusion subcategory ${\mathcal {D}}$ .

For a fusion category $\mathcal {C}$ , the fusion subcategory ${\mathcal {C}_{\mathrm {ad}}}$ represents the trivial component in the universal grading of $\mathcal {C}$ (see [Reference Gelaki and Nikshych15]). Alternatively, ${\mathcal {C}_{\mathrm {ad}}}$ is the smallest fusion subcategory of $\mathcal {C}$ which contains all the objects $X\otimes X^*$ , with $X\in \mathrm {Irr}(\mathcal {C})$ .

A grading of a fusion category $\mathcal {C}$ by a group G is defined by a map $\deg : \mathrm {Irr}(\mathcal {C}) \rightarrow G$ with the property that for any simple objects $X, Y, Z $ of $\mathcal {C}$ , if $X\otimes Y$ contains Z, then $\deg (Z) =\deg (X) \deg (Y)$ . The term ‘grading’ also refers to the decomposition $\mathcal {C}=\bigoplus _{g\in G} \mathcal {C}_g$ , where $\mathcal {C}_g $ is the full additive subcategory generated by the simple objects of degree $g \in G$ .

The subcategory $\mathcal {C}_1$ corresponding to $g =1$ is called the trivial component of the grading and is a fusion subcategory. A grading is trivial if $\mathcal {C}_1=\mathcal {C}$ , and faithful if the map $\deg : \mathrm {Irr}(\mathcal {C}) \rightarrow G$ is surjective. The set of all possible gradings of $\mathcal {C}$ by G depends functorially on G, leading to the concept of a universal grading. As shown in [Reference Gelaki and Nikshych15, Section 3.2], such a grading always exists, and the associated group is denoted by $U(\mathcal {C})$ . The following results are proven in [Reference Gelaki and Nikshych15, Section 3.2]: (see [Reference Gelaki and Nikshych15, (Lemma 3.4, Theorem 3.5, and Proposition 3.9)].

1. (1) Any faithful grading of $\mathcal {C}$ by a group G is determined by a surjective group homomorphism $\pi :U(\mathcal {C})\rightarrow G$ .

2. (2) The universal grading $\deg : \mathrm {Irr}(\mathcal {C}) \rightarrow U(\mathcal {C})$ is faithful.

3. (3) The trivial component of the universal grading equals ${\mathcal {C}_{\mathrm {ad}}}$ .

4. (4) Any fusion subcategory ${\mathcal {D}}\subseteq \mathcal {C}$ that contains ${\mathcal {C}_{\mathrm {ad}}}$ can be written as form ${\mathcal {D}}=\bigoplus _{g\in G}\mathcal {C}_g$ for some subgroup $G\leq U(\mathcal {C})$ .

It follows that $\mathrm {FPdim}(\mathcal {C})=|U(\mathcal {C})|{\mathrm {FPdim}({\mathcal {C}_{\mathrm {ad}}})}$ .

Let $\mathcal {C}$ be a fusion category. Following [Reference Gelaki and Nikshych15], we define a sequence of subcategories: $\mathcal {C}^{(0)}=\mathcal {C}$ , $\mathcal {C}^{(1)}={\mathcal {C}_{\mathrm {ad}}}$ , and for all $n\geq 1$ , $\mathcal {C}^{(n)}=\mathcal {C}^{(n-1)}_{ad}$ .

The descending sequence

$$ \begin{align*}\mathcal{C}=\mathcal{C}^{(0)}\supseteq \mathcal{C}^{(1)}\supseteq \dots \supseteq \mathcal{C}^{(n)} \supseteq \dots \end{align*} $$

is called the upper central series. A fusion category $\mathcal {C}$ is said to be nilpotent if its upper central eventually reaches $\mathrm {Vec}$ (the category of finite dimensional $\mathbb {C}$ -vector spaces), meaning that $\mathcal {C}^{(n)}= \mathrm {Vec}$ for some n. The smallest such n is called the nilpotency class of  $\mathcal {C}$ .

3 Restriction of characters of Grothendieck rings

Throughout this section, let $\mathcal {C}$ be a fusion category with a commutative Grothendieck ring, $K(\mathcal {C})$ . Then as above, ${\mathrm {CF}(\mathcal {C})}$ is a commutative semisimple algebra isomorphic to $K(\mathcal {C})\otimes _{\mathbb {Z}}\mathbb {C}$ .

As above, let $\mathcal J^{\mathcal {C}}:= \{0, \dots , m\}$ represent the set of indices corresponding to the conjugacy classes of $\mathcal {C}$ . It is clear that the rank of $\mathcal {C}$ is $\operatorname {\mathrm {{rank}}}(\mathcal {C}) = |\mathcal J^{\mathcal {C}}| = m + 1$ .

Suppose ${\mathcal {D}}$ is a fusion subcategory of $\mathcal {C}$ . Note that the Grothendieck ring $K({\mathcal {D}})$ is also commutative, and, as in the case above, $K({\mathcal {D}})\simeq \mathrm {{CF}}({\mathcal {D}})$ as algebras. We let ${\mathcal J^{{\mathcal {D}}}:= \{0, \dots , m'\}}$ represent the set of indices corresponding to the conjugacy classes of ${\mathcal {D}}$ . As above, it is clear that the rank of ${\mathcal {D}}$ is $\operatorname {\mathrm {{rank}}}({\mathcal {D}}) = |\mathcal J^{\mathcal {D}}| = m' + 1$ .

For any object $V \in \mathcal {C}$ , following [Reference Shimizu22, Sect. 4.3], consider the end

$$ \begin{align*}{\bar{\mathrm Z}}: \mathcal{C}\rightarrow \mathcal{C},\;\; {\bar{\mathrm Z}}(V):=\int_{X\in {\mathcal{D}}} X\otimes V\otimes X^*. \end{align*} $$

Let $\bar {\pi }_{V;\; X}:{\bar {\mathrm Z}}(V) \rightarrow X\otimes V\otimes X^*$ denote the universal dinatural maps that define this end. By the universal property of ${\bar {\mathrm Z}}$ , for any $V\in \mathcal {C}$ , there exists a unique canonical map in $\mathcal {C}$ :

(3.1) $$ \begin{align} Z(V)\xrightarrow{q_V}{\bar{\mathrm Z}}(V) \end{align} $$

such that $\bar {\pi }_{V;\; X}\circ q_V={\pi }_{V;\; X}$ for every object X in ${\mathcal {D}}$ .

In [Reference Burciu3, Appendix], we demonstrated that q is a morphism of Hopf comonads. The map $q_{\mathbf {1}}:Z(\mathbf {1})\rightarrow {{\bar {\mathrm Z}}(\mathbf {1})}$ induces two associated maps:

$$ \begin{align*}\pi_f=q_{\mathbf{1}}^*:{\mathrm{CF}({\mathcal{D}})} \rightarrow {\mathrm{CF}(\mathcal{C})}, \;\chi\mapsto \chi \circ q_{\mathbf{1}}, \end{align*} $$

$$ \begin{align*}{\pi_e}={q_{\mathbf{1}}}_*:\mathrm{ {CE}(\mathcal{C})} \rightarrow \mathrm{ {CE}({\mathcal{D}})}, \;z\mapsto q_{\mathbf{1}} \circ z. \end{align*} $$

Furthermore, by [Reference Burciu3, Lemma 7.7], it is known that $q_{\mathbf {1}}^*$ is a monomorphism and ${q_{\mathbf {1}}}_*$ is an epimorphism. Both maps are, in fact, $\mathbb {C}$ -algebra homomorphisms.

Let, $\{F_j\}_{0 \leq j \leq m}$ be the (central) primitive idempotents of ${\mathrm {CF}(\mathcal {C})}$ , and let $\{\mathcal {C}^j\}_{0\leq j \leq m}$ be their associated conjugacy classes. For each (central) primitive idempotent $F_j\in {\mathrm {CF}(\mathcal {C})}$ , let $\mu _j:{\mathrm {CF}(\mathcal {C})}\rightarrow \mathbb {C}$ denote its associated linear character. Similarly, let $\{G_t\}_{0 \leq t \leq m'}$ be the (central) primitive idempotents of $\mathrm {{CF}}({\mathcal {D}})$ , with $\{{\mathcal {D}}^t\}_{0\leq t \leq m'}$ being their associated conjugacy classes. Additionally, let $\nu _t: {\mathrm {CF}({\mathcal {D}})} \to \mathbb {C}$ denote the characters associated with these idempotents.

Using the map $\pi _f$ , we can identify ${\mathrm {CF}({\mathcal {D}})}$ as subring of ${\mathrm {CF}(\mathcal {C})}$ .

From the proof of [Reference Burciu5, Theorem 6.12], for all $\alpha \in {\mathrm {CF}(\mathcal {C})}$ , we have

(3.2) $$ \begin{align} \mu_j(\alpha)=\langle \alpha, \frac{C_j}{\dim(\mathcal{C}^j)}\rangle. \end{align} $$

We say that two characters $\mu _j, \mu _{j'}:{\mathrm {CF}(\mathcal {C})}\rightarrow \mathbb {C}$ are equivalent or in the same class, (denoted $\mu _j \simeq \mu _{j'}$ ) with respect to ${\mathcal {D}}$ if their restrictions to $\mathrm {{CF}}({\mathcal {D}})$ are the same.

The set of equivalence classes is denoted by $\mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ , and it is referred to as the set of classes of $\mathcal {C}$ with respect to ${\mathcal {D}}$ . This set is in bijection with the set of the characters of $\nu _t:{\mathrm {CF}({\mathcal {D}})}\rightarrow \mathbb {C}$ with $0\leq t\leq m'$ .

Moreover, an element $C\in \mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ consists of all the indices j such that the restriction $\mu _j\downarrow _{{\mathrm {CF}({\mathcal {D}})}}$ corresponds to the same character in ${\mathrm {CF}({\mathcal {D}})}$ , denoted as $\rho _C:=\nu _s$ for some ${s\leq m'}$ . Thus, we use $\rho _C$ to represent the restriction to ${\mathrm {CF}({\mathcal {D}})}$ of all characters $\mu _j$ belonging to some class $C\in \mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ .

As noted above, the set of character classes of $\mathcal {C}$ relative to ${\mathcal {D}}$ corresponds bijectively to the characters of ${\mathrm {CF}({\mathcal {D}})}$ . Consequently, the central primitive idempotents of ${\mathrm {CF}({\mathcal {D}})}$ are also indexed by the classes $C\in \mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ . We denote the central primitive idempotent associated with a class $C \in \mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ by $G_C \in \mathrm {{CF}}({\mathcal {D}})$ . Thus, the elements $G_C$ for $C \in \mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ form a linear basis for $\mathrm {{CF}}({\mathcal {D}})$ , consisting of the complete set of (central) primitive idempotents of the commutative $\mathbb {C}$ -algebra $\mathrm {{CF}}({\mathcal {D}})$ . For the remainder of this section, we identify the set of indices $\mathcal J^{\mathcal {D}}$ of the central primitive idempotents of $\mathrm {{CF}}({\mathcal {D}})$ with the elements of $\mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ . For any primitive idempotent $G_C$ of ${\mathrm {CF}({\mathcal {D}})}$ , as described in Subsection 2.1, let $\mathrm {D}_C\in \mathrm { {CE}({\mathcal {D}})}$ denote the corresponding conjugacy class sum of ${\mathcal {D}}$ associated with $G_C$ .

Proposition 3.1. Let $\mathcal {C}$ be a pivotal fusion category with a commutative Grothendieck ring, $K(\mathcal {C})$ , and let ${\mathcal {D}}$ be a fusion subcategory. With the above notation, we have

(3.3) $$ \begin{align} \pi_f(G_C)=\sum_{j \in C}F_j \end{align} $$

for any class C.

Recall the canonical pairing between ${\mathrm {CF}(\mathcal {C})}$ and $\mathrm { {CE}(\mathcal {C})}$ , which we denote as $\langle ,\;\rangle _{\mathcal {C}}$ . For the fusion subcategory ${\mathcal {D}}\subseteq \mathcal {C}$ , there is an analogous pairing, written here as $\langle ,\;\rangle _{\mathcal {D}}$ . The proof of the following lemma follows directly from the definitions of the two canonical pairings.

Lemma 3.2. The two canonical pairings on the fusion categories $\mathcal {C}$ and ${\mathcal {D}}$ are compatible, satisfying the relation

(3.4) $$ \begin{align} \langle\pi_f(\chi), \;z\rangle_{\mathcal{C}}=\langle \alpha, \pi_e(z)\rangle_{\mathcal{D}} \end{align} $$

for all $\chi \in {\mathrm {CF}({\mathcal {D}})}$ and $z\in \mathrm { {CE}(\mathcal {C})}$ .

3.1 On the projection of conjugacy class sums

Theorem 3.3. Let $\mathcal {C}$ be a pivotal fusion category with a commutative Grothendieck ring, and ${\mathcal {D}}$ be a fusion subcategory of $\mathcal {C}$ . Let $j\in \mathcal {J}$ be any index and suppose that $j\in C$ for some $C\in \mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ . Then

(3.5) $$ \begin{align} \pi_e({\frac{C_j}{\dim(\mathcal{C}^j)}})=\frac{\mathrm{D}_C}{\dim({\mathcal{D}}^C)}. \end{align} $$

Proof. We can write

$$ \begin{align*}\pi_e({\frac{C_j}{\dim(\mathcal{C}^j)}})=\sum_{C_1 \in \mathrm{Cl}(\mathcal{C}:{\mathcal{D}})}\alpha_{jC_1}\mathrm{D}_{C_1} \end{align*} $$

for some scalars $\alpha _{jC_1}\in \mathbb {C}$ .

Let now $C_2\in \mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ be an arbitrary class, and let $\chi =G_{C_2}$ and $z=C_j$ . By Lemma 3.2, we have

$$ \begin{align*}\langle \sum_{s \in C_2} F_s, {\frac{C_j}{\dim(\mathcal{C}^j)}} \rangle_{ _{\mathcal{C}}}=\langle G_{C_2}, \pi_e({\frac{C_j}{\dim(\mathcal{C}^j)}}) \rangle_{_{\mathcal{D}}}, \end{align*} $$

which in turn can be written as

$$ \begin{align*}\langle \sum_{s \in C_2} F_s, {\frac{C_j}{\dim(\mathcal{C}^j)}} \rangle_{ _{\mathcal{C}}}=\langle G_{C_2}, \sum_{C_1\in \mathrm{Cl}(\mathcal{C}:{\mathcal{D}})}\alpha_{jC_1}\mathrm{D}_{C_1}\rangle_{_{\mathcal{D}}}. \end{align*} $$

The right-hand side of the last above equation, using the dual bases form Equation 2.5 for ${\mathcal {D}}$ , can be written as

$$ \begin{align*}\langle G_{C_2}, \sum_{C_1 \in \mathrm{Cl}(\mathcal{C}:{\mathcal{D}})}\alpha_{jC_1}\mathrm{D}_{C_1}\rangle_{ _{\mathcal{D}}}=\alpha_{jC_2}\dim({\mathcal{D}}^{C_2}).\end{align*} $$

However, using also the dual bases form Equation 2.5, for $\mathcal {C}$ this time, the left-hand side equals $\delta _{j\in C_2}$ .

Thus, we find that $\alpha _{jC_2}\dim ({\mathcal {D}}^{C_2})=\delta _{j\in C_2}$ for all character classes $C_2\in \mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ . Since $j\in C$ , this gives that $\alpha _{jC} = \frac {1}{\dim ({\mathcal {D}}^C)}$ and $\alpha _{jC_2}=0$ for all $C_2\neq C$ . This establishes Equation (3.5).

For a fusion category $\mathcal {C}$ , we define the natural coend projection $\epsilon _{\mathcal {C}}:A\rightarrow \mathbf {1}\in {\mathrm {CF}(\mathcal {C})}$ as the map $A\xrightarrow {\pi _{\mathbf {1}, \mathbf {1}}}\mathbf {1} \otimes \mathbf {1}^*\simeq \mathbf {1}$ . This projection is referred to as $\epsilon _{\mathbf {1}}$ in [Reference Shimizu22], but we use the notation $\epsilon _{\mathcal {C}}$ to emphasize the category $\mathcal {C}$ .

Recall that in the previous section, we defined $\chi _0\in {\mathrm {CF}(\mathcal {C})}$ as the irreducible character corresponding to the tensor unit $\mathbf {1}_{\mathcal {C}}$ of $\mathcal {C}$ . Without loss of generality, we can also assume that $\alpha _0\in {\mathrm {CF}({\mathcal {D}})}$ is the character associated to the unit object $\mathbf {1}_{\mathcal {D}}$ of ${\mathcal {D}}$ . It is clear that from its definition that $\pi _f(\alpha _0)=\chi _0$ .

Lemma 3.4. Let ${\mathcal {D}}\subseteq \mathcal {C}$ be a fusion subcategory of a pivotal fusion category $\mathcal {C}$ . Under the above notation, we have

(3.6) $$ \begin{align} \epsilon_{\mathcal{D}}\circ {q_{\mathbf{1}}}=\epsilon_{\mathcal{C}} \end{align} $$

and

(3.7) $$ \begin{align} \pi_e(\Lambda_{\mathcal{C}})=\Lambda_{\mathcal{D}}. \end{align} $$

Proof. From the naturality of ${q_{\mathbf {1}}}$ , we can conclude that $\epsilon _{\mathcal {D}}\circ {q_{\mathbf {1}}}=\epsilon _{\mathcal {C}}$ , since $\epsilon _{\mathcal {C}}:A\rightarrow \mathbf {1}$ serves as the universal end projection $A\xrightarrow {\pi _{\mathbf {1}, \mathbf {1}}}\mathbf {1} \otimes \mathbf {1}^*\simeq \mathbf {1}$ .

For the second assertion, we first show that $\pi _e(\Lambda _{\mathcal {C}})\neq 0$ . According to [Reference Shimizu22, Lemma 6.1], the corresponding idempotent of $\chi _0$ is $E_0=\Lambda _{\mathcal {C}}$ . By applying Lemma 3.2, we have

$$ \begin{align*}1=\langle\pi_f(\alpha_0), E_0\rangle=\langle\alpha_0, \pi_e(\Lambda_{\mathcal{C}})\rangle, \end{align*} $$

which implies that $\pi _e(\Lambda _{\mathcal {C}})\neq 0$ . Moreover, for any $z \in \mathrm { {CE}(\mathcal {C})}$ , we find

$$ \begin{align*}\epsilon_{\mathcal{C}}(z)\pi_e(\Lambda_{\mathcal{C}})=\pi_e(z\Lambda_{\mathcal{C}})=\pi_e(z)\pi_e(\Lambda_{\mathcal{C}}). \end{align*} $$

Note that $\epsilon _{\mathcal {D}}(\pi (z))=\epsilon _{\mathcal {C}}(z)$ , and since $\pi _e$ is surjective, it follows that

$$ \begin{align*}w\pi_e(\Lambda_{\mathcal{C}})=\epsilon_{\mathcal{D}}(w)\pi_e(\Lambda_{\mathcal{C}}) \end{align*} $$

for any $w\in \mathrm { {CE}({\mathcal {D}})}$ . Given that $\pi _e(\Lambda _{\mathcal {C}})\neq 0$ , it follows $\pi _e(\Lambda _{\mathcal {C}})=\Lambda _{\mathcal {D}}$ .

4 On the dimensions of conjugacy classes

For any weakly integral fusion category, whose Grothendieck group $K(\mathcal {C})$ is commutative, the following Harada type identity holds, as established in [Reference Burciu6, Theorem 1]:

(4.1) $$ \begin{align} (\prod_{j=0}^m \frac{C_j}{\mathrm{FPdim}(\mathcal{C}^j)})^2=\frac{\mathrm{FPdim}({\mathcal{C}_{\mathrm{pt}}})}{\mathrm{FPdim}(\mathcal{C})}\big(\sum_{j\in \mathcal J_{{\mathcal{C}_{\mathrm{pt}}}}}C_j\big). \end{align} $$

Recall that the support of ${\mathcal {D}}$ in $\mathcal {C}$ is the subset of indices of $\mathcal J_{{\mathcal {D}}}\subseteq \mathcal J$ such that $\lambda _{{\mathcal {D}}}=\sum _{j \in \mathcal J_{\mathcal {D}}}F_j$ .

Since the elements $\{C_j\}$ form a linear basis for $\mathrm { {CE}(\mathcal {C})}$ , we can express the product of two basis elements as

(4.2) $$ \begin{align} C_iC_j=\sum_{k=0}^m c^k_{ij}C_k, \end{align} $$

where $c^k_{ij}\in \mathbb C$ are the structure constants of $\mathcal {C}$ . It is known that these constants are in fact integers in the case $\mathcal {C}=\mathrm {Rep}(G)$ for a finite group G. Burnside provided a formula relating these structure constants to the values of irreducible characters; see [Reference Isaacs16, pp. 45]. A similar result for the structure constants of a pivotal fusion category with a commutative Grothendieck ring was presented in [Reference Burciu4, Theorem 1.1].

A braided fusion category with a spherical structure is known as a premodular fusion category. Note that every weakly integral braided fusion category is premodular, by [Reference Etingof, Nikshych and Ostrik14, Prop 8.23].

By [Reference Burciu4, Theorem 1.2], we have

(4.3) $$ \begin{align} \dim(\mathcal{C}^k)c^k_{ij}\in \mathbb Z_{\geq 0} \end{align} $$

for any structure constant $c^k_{ij}$ in a weakly integral braided fusion category.

4.1 Fusion categories with rational structure constants

Remark 4.1. We note that in any weakly integral fusion category, we have $\mathrm {FPdim}(\mathcal {C}^j)\in \mathbb {Z}$ . First, it is easy to see that for any fusion category $\mathcal {C}$ , the forgetful images $F(\mathcal {C}^j)$ of the conjugacy classes are objects of ${\mathcal {C}_{\mathrm {ad}}}$ . Indeed, as shown in [Reference Etingof, Nikshych and Ostrik14, Prop. 5.4], we have that $F(R(\mathbf {1}))= \bigoplus _{i=0}^m V_i \otimes V_{i^*}$ . However, from above, we have $R(\mathbf {1})=\oplus _{j=0}^m\mathcal {C}^j$ , see §2.1, which gives that each $F(\mathcal {C}^j)$ is a direct summand of $\oplus _{i=0}^m V_i\otimes V_{i^*}$ . By [Reference Etingof, Nikshych and Ostrik14, Prop 8.27], the Frobenius-Perron dimensions of the simple objects in $\mathcal {C}_{ad}$ are integers. Since $F(\mathcal {C}^j)$ is a sum of simple objects of the adjoint subcategory ${\mathcal {C}_{\mathrm {ad}}}$ , it follows that $\mathrm {FPdim}(\mathcal {C}^j)\in \mathbb {Z}$ .

Theorem 4.2. Let $\mathcal {C}$ be a weakly integral fusion category, and suppose that all its structure constants $c^k_{ij}$ are rational numbers. Let $D\in \mathbb Z$ be such that $Dc^k_{ij}\in \mathbb Z$ for all $i, j, k$ . Then

1. (1) $\large {\frac {D^{2m+1} {\widehat {P}}^{\;2}\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})}{\mathrm {FPdim}(\mathcal {C})}\in \mathbb Z}$ .

2. (2) $d_i\mid D\;\mathrm {FPdim}(\mathcal {C})$

3. (3) $d_i\mid D^{2m+2} {\widehat {P}}^{\;2}\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})$ .

Proof. Let ${\widehat {P}}:=\prod _{j=0}^m\mathrm {FPdim}(\mathcal {C}^j)$ . By Remark 4.1, we know that ${\widehat {P}}\in \mathbb Z$ .

Multiplying Equation (4.2) by D, we obtain

$$ \begin{align*}D C_iC_j=\sum_{k=0}^m (D c^k_{ij})C_k, \end{align*} $$

with $D c^k_{ij}\in \mathbb Z$ , by assumption. By induction, for any r-class sums, the product $D^{r-1}C_{i_1}C_{i_2}\dots C_{i_r}$ yields integral coefficients for each $C_k$ when expressed as a linear combination of conjugacy class sums.

Next, by multiplying both sides of Equation(4.1) by $D^{2m+1}$ , we obtain

(4.4) $$ \begin{align} D^{2m+1}(\prod_{j=0}^m {C_j})^2=\frac{D^{2m+1}{\widehat{P}}^{2} \mathrm{FPdim}({\mathcal{C}_{\mathrm{pt}}})}{\mathrm{FPdim}(\mathcal{C})}\big(\sum_{j\in \mathcal J_{{\mathcal{C}_{\mathrm{pt}}}}}C_j\big). \end{align} $$

In the product $D^{2m+1}(\prod _{j=0}^m {C_j})^2$ , all $C_j$ , where $j\in \mathcal J_{\mathcal {C}_{\mathrm {pt}}}$ , have integral coefficients. Thus, it follows that the coefficient of $C_j$ on the right hand side is also integral, which implies that $\frac {D^{2m+1}{\widehat {P}}^{\;2}\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})}{\mathrm {FPdim}(\mathcal {C})}\in \mathbb Z$ . This completes the proof of the first item.

The second item is a direct consequence of [Reference Burciu and Palcoux2, Theorem 4.7]. The third item follows directly from the first two divisibility results.

4.1.1 Proof of Theorem 1.1

Proof. Since $\mathcal {C}$ is weakly integral braided, we have that $\dim (\mathcal {C}^k)c^k_{ij}\in \mathbb Z_{\geq 0}$ by Equation (4.3). Thus, one can apply Theorem 4.2 for $D={\widehat {P}}$ .

Corollary 4.3. Let $\mathcal {C}$ be a weakly integral braided fusion category. Then $d_i\mid {\widehat {P}}^{\;2\operatorname {\mathrm {{rank}}}(\mathcal {C})+2}\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})$ for all $0\leq i\leq m$ .

Lemma 4.4. Let $\mathcal {C}$ be a weakly integral fusion category, and suppose that all its structure constants $c^k_{ij}$ are rational numbers. Let ${\mathcal {D}}$ be a fusion subcategory of ${\mathcal {D}}$ . Then ${\mathcal {D}}$ has also rational structure constants.

Proof. Recall the map $\pi _e:\mathrm { {CE}(\mathcal {C})}\rightarrow \mathrm { {CE}({\mathcal {D}})}$ from Section 3. Let ${\mathcal {D}}_E$ and ${\mathcal {D}}_F$ be two conjugacy classes of ${\mathcal {D}}$ with $E, F\in \mathrm {Cl}(\mathcal {C}:{\mathcal {D}})$ . Since $\pi _e$ is a surjective map, as in Equation (3.5), there are indices $s\in E,\; t\in F$ such that $ \pi _e(\mathrm {C}_{t})=\frac {\dim (\mathcal {C}^{t})}{\dim ({\mathcal {D}}^E)}\mathrm {D}_E $ and $ \pi _e(\mathrm {C}_{s})=\frac {\dim (\mathcal {C}^{s})}{\dim ({\mathcal {D}}^F)}\mathrm {D}_F. $ Then

$$ \begin{align*}\mathrm{D}_E\mathrm{D}_F=\frac{\dim({\mathcal{D}}^E)}{\dim(\mathcal{C}^{t})}\frac{\dim({\mathcal{D}}^F)}{\dim(\mathcal{C}^{s})}\pi_e(\mathrm{C}_{t})\pi_e(\mathrm{C}_{s}). \end{align*} $$

However, since $\pi _e$ is an algebra homomorphism, one has

$$ \begin{align*} \pi_e(\mathrm{C}_{t})\pi_e(\mathrm{C}_{s}) &= \pi_e(\mathrm{C}_{t}\mathrm{C}_{s})=\pi_e(\sum_{k=0}^m c^k_{ts}\mathrm{C}_k) \\ &= \sum_{k=0}^m c^k_{ts}\pi_e(\mathrm{C}_k) \\ &= \sum_{C\in \mathrm{Cl}(\mathcal{C}:{\mathcal{D}})}\big(\sum_{k\in C} c^k_{ts}\pi_e(\mathrm{C}_k)\big) \\ &= \sum_{C\in \mathrm{Cl}(\mathcal{C}:{\mathcal{D}})}\big(\sum_{k\in C}c^k_{ts}\frac{\dim(\mathcal{C}^{k})}{\dim({\mathcal{D}}^{C})}\big) \mathrm{D}_{C}\\ &= \sum_{C\in \mathrm{Cl}(\mathcal{C}:{\mathcal{D}})}\big(\sum_{k\in C}c^k_{ts}\dim(\mathcal{C}^{k})\big)\frac{ \mathrm{D}_{C}}{\dim({\mathcal{D}}^{C})}. \end{align*} $$

Therefore,

$$ \begin{align*} \mathrm{D}_E\mathrm{D}_F = \frac{\dim({\mathcal{D}}^E)}{\dim(\mathcal{C}^{t})} \frac{\dim({\mathcal{D}}^F)}{\dim(\mathcal{C}^{s})}\sum_{C\in \mathrm{Cl}(\mathcal{C}:{\mathcal{D}})}\big(\sum_{k\in C}c^k_{ts}\dim(\mathcal{C}^{k})\big)\frac{ \mathrm{D}_{C}}{\dim({\mathcal{D}}^{C})}, \end{align*} $$

which shows that the coefficient of each conjugacy class sum $ \mathrm {D}_C$ is a rational number.

Proposition 4.5. Let $\mathcal {C}$ be a weakly integral fusion category with a commutative Grothendieck ring and suppose that all its structure constants are integral (i.e., $c^k_{ij}\in \mathbb Z$ ). With above notation, one has

(4.5) $$ \begin{align} \frac{\mathrm{FPdim}({\mathcal{C}_{\mathrm{pt}}})}{\mathrm{FPdim}(\mathcal{C})}\big[\prod_{j=0}^m\mathrm{FPdim}(\mathcal{C}^j)\big]^{2}\in \mathbb Z. \end{align} $$

Moreover, if $U(\mathcal {C})$ has odd order, then we have

(4.6) $$ \begin{align} \frac{\mathrm{FPdim}({\mathcal{C}_{\mathrm{pt}}})}{\mathrm{FPdim}(\mathcal{C})}\big[\prod_{j=0}^m\mathrm{FPdim}(\mathcal{C}^j)\big]\in \mathbb Z. \end{align} $$

Proof. If $\mathcal {C}$ has integral structure constants, then we can take $D=1$ since the coefficient of $C_0$ in the left-hand side of Equation (4.1) is already an integer. Therefore, in this case, there is no need to multiply Equation (4.1) by ${\widehat {P}}^{\;2m+1}$ , and we directly obtain

$$ \begin{align*}\frac{\mathrm{FPdim}({\mathcal{C}_{\mathrm{pt}}})}{\mathrm{FPdim}(\mathcal{C})}\big[\prod_{j=0}^m\mathrm{FPdim}(\mathcal{C}^j)\big]^{2}\in \mathbb Z. \end{align*} $$

Using the dual version of [Reference Burciu and Palcoux1, Corollary 4.10], in the case in which $U(\mathcal {C})$ has odd order, we also have the second divisibility result.

A well-known result regarding the prime factors of the sizes of conjugacy classes in finite groups can be found, for example, in [Reference Chillag and Herzog7, Proposition 4]. Let G be a finite group and p a prime divisor of G. Then, p does not divide the size of any conjugacy class of G if and only if the Sylow p-subgroup of G is contained in its center. In this case, Schur–Zassenhaus theorem (see [Reference Robinson21]) ensures that $G\simeq H\times K$ , where H is its Sylow p-subgroup and K is another subgroup of G. In particular, this shows that $p^{\alpha }$ divides the order of the abelian group $G/G'$ (i.e., dimension of $\mathrm {Rep}(G)_{pt}$ ).

As a consequence of Theorem 1.1, a weaker version of this result can be proven for weakly integral braided fusion categories in Theorem 1.2.

4.4 Proof of Theorem 1.4

Proof. We apply Proposition 1.3 repeatedly for each prime that does not divide the dimension of any simple object in $\mathcal {C}$ . Suppose $\mathrm {FPdim}(\mathcal {C}) = p_1^{\alpha _1} \dots p_n^{\alpha _n}$ . Without loss of generality, assume that the primes $p_i$ for $1 \leq i \leq r$ do not divide the dimension of any simple object in $\mathcal {C}$ .

Applying Proposition 1.3 with $p = p_1$ , we obtain a decomposition $\mathcal {C} = {\mathcal {D}}_1 \boxtimes {\mathcal {E}}_1$ , where ${\mathcal {D}}_1$ is a pointed modular fusion category with $\mathrm {FPdim}({\mathcal {D}}_1) = p_1^{\alpha _1}$ . Since $\mathrm {FPdim}({\mathcal {E}}_1) = p_2^{\alpha _2} \dots p_n^{\alpha _n}$ , we can apply Proposition 1.3 again with $p = p_2$ , yielding ${\mathcal {E}}_1 = {\mathcal {D}}_2 \boxtimes {\mathcal {E}}_2$ , where ${\mathcal {D}}_2$ is a pointed modular fusion category with $\mathrm {FPdim}({\mathcal {D}}_2) = p_2^{\alpha _2}$ .

Continuing in this way for each prime $p_i$ with $1 \leq i \leq r$ , we obtain

$$ \begin{align*}\mathcal{C}\simeq {\mathcal{D}}_1\boxtimes \dots \boxtimes{\mathcal{D}}_r\boxtimes {\mathcal{E}}_r. \end{align*} $$

Thus, we set ${\mathcal {D}}:={\mathcal {D}}_1\boxtimes \dots \boxtimes {\mathcal {D}}_r$ and ${\mathcal {E}}:={\mathcal {E}}_r$ .

Recall that a fusion category is called perfect if ${\mathcal {C}_{\mathrm {pt}}}=\mathrm {Vec}$ .

Corollary 4.8. For any weakly integral perfect modular fusion category $\mathcal {C}$ , if p is a prime divisor of $\mathrm {FPdim}(\mathcal {C})$ , then there exists a simple object in $\mathcal {C}$ whose squared dimension is divisible by p.

5 Proof of Theorem 1.5

We would like to thank A. Moreto for suggesting the proof of the following result:

Proposition 5.1. Let $\mathcal {C}$ be a pivotal fusion category with a commutative Grothendieck ring $K(\mathcal {C})$ . Then for any character class $C \in \mathrm {Cl}(\mathcal {C}:{\mathcal {C}_{\mathrm {pt}}})$ , we have

(5.1) $$ \begin{align} \sum_{l \in C} \dim(\mathcal{C}^l) = \frac{\dim(\mathcal{C})}{\dim({\mathcal{C}_{\mathrm{pt}}})}. \end{align} $$

Proof. Since ${\mathcal {C}_{\mathrm {pt}}}\subseteq \mathcal {C}$ , by [Reference Burciu3, Lemma 7.5, item 1)], there exists a surjective homomorphism of $\mathbb {C}$ -algebras:

$$ \begin{align*}\pi_e: \mathrm{ {CE}(\mathcal{C})} \rightarrow \mathrm{CE}({\mathcal{C}_{\mathrm{pt}}}), \end{align*} $$

as discussed in Section 3.

Let ${\mathcal {C}_{\mathrm {pt}}}:=\mathrm {Vec}_G^\omega $ for some abelian group G and some $\omega \in H^3(G, \mathbb {C}^*)$ . It follows from Example 2.2, and we have $\mathrm {CE}({\mathcal {C}_{\mathrm {pt}}})= \mathbb {C}[G]$ . Note that the conjugacy classes of $\mathrm {Vec}_G^\omega $ correspond to the elements of G.

According to Theorem 3.3, for any character class $C \in \mathrm {Cl}(\mathcal {C}:{\mathcal {C}_{\mathrm {pt}}})$ , there exists a group element $g_C\in G$ such that

$$ \begin{align*}\pi_e(C_j)= \dim(\mathcal{C}^j) g_C, \;\text{for all}\;j\in C. \end{align*} $$

This establishes a bijection $C\mapsto g_C$ between the sets $\mathrm {Cl}(\mathcal {C}:{\mathcal {C}_{\mathrm {pt}}})$ and G. Additionally, by Lemma 3.4, we have

(5.2) $$ \begin{align} \pi_e( \widehat{\Lambda}_{\mathcal{C}}) =\frac{\dim(\mathcal{C})}{\dim({\mathcal{C}_{\mathrm{pt}}})} \widehat{\Lambda}_{\mathcal{C}_{\mathrm{pt}}}. \end{align} $$

Since $\mathrm {CE}({\mathcal {C}_{\mathrm {pt}}})= \mathbb {C}[G]$ , it follows that $\widehat {\Lambda }_{\mathcal {C}_{\mathrm {pt}}}=\sum _{g \in G} g$ . Recall that, according to [Reference Burciu3, Proposition 4.1] (applied for ${\mathcal {D}}=\mathcal {C}$ ), we have

$$ \begin{align*}\widehat{\Lambda}_{\mathcal{C}}=\sum_{j\in \mathcal J} C_j. \end{align*} $$

Counting the multiplicity of $g_C$ in the left-hand side of Equation (5.2), we find

$$ \begin{align*}\sum_{l \in C} \dim(\mathcal{C}^l) = \frac{\dim(\mathcal{C})}{\dim({\mathcal{C}_{\mathrm{pt}}})}.\\[-45pt] \end{align*} $$

Note that the proof of the above proposition implies the following:

Corollary 5.2. For any pseudo-unitary fusion category $\mathcal {C}$ with a commutative Grothendieck ring $K(\mathcal {C})$ , we have

(5.3) $$ \begin{align} \mathrm{FPdim}(\mathcal{C}^j)\leq \frac{\mathrm{FPdim}(\mathcal{C})}{{\mathrm{FPdim}({\mathcal{C}_{\mathrm{pt}}})}}, \end{align} $$

for any conjugacy class $\mathcal {C}^j$ of $\mathcal {C}$ .

Recall that $\mathcal {C}$ is called a p-fusion category if $\mathrm {FPdim}(\mathcal {C})=p^n$ for some $n\geq 1$ .

Corollary 5.4. For any p-fusion category, one has

$$ \begin{align*}\mathrm{FPdim}(\mathcal{C}^j)\mid \frac{\mathrm{FPdim}(\mathcal{C})}{{\mathrm{FPdim}({\mathcal{C}_{\mathrm{pt}}})}}. \end{align*} $$

5.1 Proof of Theorem 1.5

Proof. As above, $\mathcal {C}$ is weakly integral since it is nilpotent. Furthermore, any braided nilpotent fusion category can be uniquely decomposed into a Deligne product of p-categories, as established in [Reference Drinfeld, Gelaki, Nikshych and Ostrik12, Theorem 1.1]. Consequently, the divisibility property follows from Corollary 5.4. Indeed, if $\mathcal {C}\simeq \prod _{p\mid \mathrm {FPdim}(\mathcal {C})} \mathcal {C}_p$ with each $\mathcal {C}_p$ being a p-fusion category, then it is straightforward to see that ${\mathcal {Z}(\mathcal {C})}\simeq \prod _{p\mid \mathrm {FPdim}(\mathcal {C})} \mathcal {Z}(\mathcal {C}_p)$ . Thus, each conjugacy class of $\mathcal {C}$ is a Deligne product of a conjugacy class of each $\mathcal {C}_p$ . Moreover, ${\mathcal {C}_{\mathrm {pt}}}\simeq \prod _{p\mid \mathrm {FPdim}(\mathcal {C})} (\mathcal {C}_p)_{pt}$ , and the proof of the first part of the Theorem follows from Corollary 5.4.

For the second part, by Theorem (1.1), we have that

(5.4) $$ \begin{align} \frac{{\mathrm{FPdim}({\mathcal{C}_{\mathrm{pt}}})}{\widehat{P}}^{2\operatorname{\mathrm{{rank}}}(\mathcal{C})+1}}{{\mathrm{FPdim}(\mathcal{C})}}\in \mathbb Z, \end{align} $$

where ${\widehat {P}}:=\prod _{j=0}^m \dim (\mathcal {C}^j)$ .

It follows from here that $\mathrm {FPdim}(\mathcal {C})/{\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})}$ divides ${\widehat {P}}^{2\operatorname {\mathrm {{rank}}}(\mathcal {C})+1}$ , and this completes the proof.

5.2 Nilpotent modular categories

Let $\mathcal {C}$ be a modular category of Frobenius-Perron dimension $\mathrm {FPdim}(\mathcal {C})=p^n$ for some prime number p. Since $\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})$ divides $ \mathrm {FPdim}(\mathcal {C})$ , it follows that $\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})=p^m$ for some $0\leq m\leq n$ .

Note that [Reference Czenky, Plavnik and Schopieray8, Lemma 2] establishes that for an odd prime number p, in any modular tensor category of dimension $p^n$ , one has that

$$ \begin{align*}2 \leq m \leq n-3. \end{align*} $$

Since p is odd, it follows that $\mathrm {FPdim}(\mathcal {C})$ is odd, and thus, $\mathcal {C}$ is an integral fusion category by [Reference Gelaki and Nikshych15, Corollary 3.11.].

Using Theorem 1.1, we show in this subsection that if the upper bound $m=n-3$ is attained, then the Frobenius-Perron dimension of simple objects of $\mathcal {C}$ are either $1$ or p. Additionally, employing a slightly improved version of the Class Equation, from [Reference Etingof, Nikshych and Ostrik14, Proposition 5.7], we show that $m\geq 2$ in the case $p=2$ , too.

Remark 5.5. In the case of a modular tensor category $\mathcal {C}$ , the Drinfeld map $f_Q:{\mathrm {CF}(\mathcal {C})}\rightarrow \mathrm { {CE}(\mathcal {C})}$ , as discussed in [Reference Shimizu22, Example 6.14], is a an algebra isomorphism. As shown in [Reference Burciu3, Section 6], this isomorphism establishes a bijection between the set of primitive idempotents $\{F_j\}$ of ${\mathrm {CF}(\mathcal {C})}$ and the set $\{E_i\}$ of primitive idempotents of $\mathrm { {CE}(\mathcal {C})}$ . Note the latter set consists of the primitive central idempotents associated to the simple objects of $\mathcal {C}$ .

Specifically, for any primitive idempotent $F_j$ of ${\mathrm {CF}(\mathcal {C})}$ , we have that $\mathrm {f}_{Q}(E_j)$ is a primitive idempotent of $\mathrm { {CE}(\mathcal {C})}$ . Therefore, we have $\mathrm {f}_{Q}(F_j)=E_i$ for some $E_i$ , yielding a bijection $j\leftrightarrow i$ . Under this bijection, the second item of [Reference Burciu3, Theorem 6.9] implies that $\dim (\mathcal {C}^j)=d_j^2$ for any $j \in \mathcal J$ .

We show the following result for an arbitrary prime p.

Lemma 5.6. Let $\mathcal {C}$ be an integral modular category of Frobenius-Perron dimension $\mathrm {FPdim}(\mathcal {C})=p^n$ for some prime number p. Suppose that $\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})=p^{n-3}$ . Then the Frobenius-Perron dimensions of the simple objects of $\mathcal {C}$ are either $1$ or p.

Proof. With the above notation, note that for all $j \in \mathcal J$ , we have $\mathrm {FPdim}(\mathcal {C}^j)=d_j^2$ , by Remark 5.5. Since $\mathrm {FPdim}(\mathcal {C}^j)\mid \frac {\mathrm {FPdim}(\mathcal {C})}{\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})} =p^3$ , it follows that $d_j^2\mid p^3$ , and therefore, $d_j\in \{1,p\}$ since $d_j\in \mathbb Z$ .

Remark 5.7. For p odd, as explained above, we have $2\leq m\leq n-3$ by [Reference Czenky, Plavnik and Schopieray8, Lemma 2]. Thus, in this case, the above lemma implies that either $ m\leq n-4$ or the Frobenius-Perron dimensions of the simple objects of $\mathcal {C}$ are either $1$ or p.

For $p=2$ , note that the proof of the above Lemma also applies in the case ${\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})}=2^{n-2}$ since, in this case, we obtain $\mathrm {FPdim}(\mathcal {C}^j)\mid \frac {\mathrm {FPdim}(\mathcal {C})}{\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})} =2^2$ .

5.2.1 On the Class Equation and $m\geq 2$

A version of the Class Equation for pivotal fusion categories was presented in [Reference Etingof, Nikshych and Ostrik14, Proposition 5.7]. In this subsection, we prove a slightly improved version of the Class Equation for pseudo-unitary fusion categories.

Lemma 5.8. Let $\mathcal {C}$ be a pseudo-unitary fusion category with a commutative Grothendieck ring. Then, we have the following analogue of the group Class Equation:

(5.5) $$ \begin{align} \mathrm{FPdim}(\mathcal{C})=\frac{\mathrm{FPdim}(\mathcal{C})}{{\mathrm{FPdim}({\mathcal{C}_{\mathrm{ad}}})}}+\sum_{j\in \mathcal J\setminus \mathcal{J}_{{\mathcal{C}_{\mathrm{ad}}}}}\mathrm{FPdim}(\mathcal{C}^j). \end{align} $$

In this equation, all summands on the right-hand side are divisors of $\mathrm {FPdim}(\mathcal {C})$ in the ring of algebraic integers.

Remark 5.9. Additionally, we note that $\frac {\mathrm {FPdim}(\mathcal {C})}{{\mathrm {FPdim}({\mathcal {C}_{\mathrm {ad}}})}}={|{U(\mathcal {C})}|}$ by [Reference Etingof, Nikshych and Ostrik14, Proposition 8.20].

Proof. For any fusion category with a commutative Grothendieck ring, we have that $R(\mathbf {1})=\oplus _{j \in \mathcal J}\mathcal {C}^j$ . Therefore, we can express the dimension as

$$ \begin{align*}\dim(\mathcal{C})=\sum_{j \in \mathcal J}\dim(\mathcal{C}^j)=\sum_{j \in {J_{ _{{\mathcal{C}_{\mathrm{ad}}}}}}}\dim(\mathcal{C}^j)+\sum_{j\in \mathcal J\setminus \mathcal{J}_{{\mathcal{C}_{\mathrm{ad}}}}}\dim(\mathcal{C}^j). \end{align*} $$

Since $\mathcal {C}$ is pseudo-unitary and has a commutative Grothendieck ring, by [Reference Burciu3, Lemma 4. 6], it follows that $\mathcal J_{{\mathcal {C}_{\mathrm {ad}}}}=\{j\in \mathcal J\;|\;\dim (\mathcal {C}^j)=1\}.$ Thus, the above relation can be written as

(5.6) $$ \begin{align} \mathrm{FPdim}(\mathcal{C})=|{J_{ _{{\mathcal{C}_{\mathrm{ad}}}}}}|+\sum_{j\in \mathcal J\setminus \mathcal{J}_{{\mathcal{C}_{\mathrm{ad}}}}}\mathrm{FPdim}(\mathcal{C}^j). \end{align} $$

However, by applying [Reference Burciu3, Equation (4.16)] for ${\mathcal {D}}={\mathcal {C}_{\mathrm {ad}}}$ , we obtain

$$ \begin{align*} \frac{\dim(\mathcal{C})}{\dim({\mathcal{C}_{\mathrm{ad}}})}=\sum_{j \in {J_{ _{{\mathcal{C}_{\mathrm{ad}}}}}}}\dim(C_j). \end{align*} $$

In the pseudo-unitary case, this can be rewritten as $|{J_{ _{{\mathcal {C}_{\mathrm {ad}}}}}}|=\frac {\mathrm {FPdim}(\mathcal {C})}{{\mathrm {FPdim}({\mathcal {C}_{\mathrm {ad}}})}}$ , which proves Equation (5.5). The assertion that $\frac {\dim (\mathcal {C})}{\dim (\mathcal {C}^j)}\in \mathbb A$ follows from [Reference Etingof, Nikshych and Ostrik14, Proposition 3.3].

Using the Class Equation established above, we prove the following:

Corollary 5.10. Let $\mathcal {C}$ be an integral modular category of Frobenius-Perron dimension $\mathrm {FPdim}(\mathcal {C})=p^n$ for $n\geq 2$ and some prime number p. Suppose that $\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})=p^{m}$ . Then $m\geq 2$

Proof. For all $j \in \mathcal J$ , note that $\mathrm {FPdim}(\mathcal {C}^j)=d_j^2$ , by Remark 5.5. It follows that $p^2\mid \mathrm {FPdim}(\mathcal {C}^j)$ if $j \notin {J_{ _{{\mathcal {C}_{\mathrm {ad}}}}}}$ since $\mathcal {C}$ is integral, and therefore, $d_j\in \mathbb {Z}$ . Considering the Class Equation,

$$ \begin{align*}\mathrm{FPdim}(\mathcal{C})={|{U(\mathcal{C})}|}+\sum_{j\in \mathcal J\setminus \mathcal{J}_{{\mathcal{C}_{\mathrm{ad}}}}}\mathrm{FPdim}(\mathcal{C}^j), \end{align*} $$

we see that since $p^2\mid \mathrm {FPdim}(\mathcal {C}^j)$ for all $j\notin \mathcal {J}_{{\mathcal {C}_{\mathrm {ad}}}}$ , it follows that $p^2\mid |U(\mathcal {C})|$ . Furthermore, since $\mathcal {C}$ is modular, we have that ${|{U(\mathcal {C})}|}={\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})}$ from [Reference Etingof, Gelaki, Nikshych and Ostrik13, Corollary 8.22.8]. Therefore, it follows that $|U(\mathcal {C})|=|\mathrm {FPdim}({\mathcal {C}_{\mathrm {pt}}})|$ , which implies $m\geq 2$ .