An Ito-Michler type theorem and the prime class graph of weakly integral braided fusion categories

Introduction

Let $G$ be a finite group and $p$ a prime divisor of $|G|$. The Itô–Michler theorem characterizes the absence of $p$ from the degrees of irreducible characters of $G$ by the existence of a normal abelian Sylow $p$-subgroup [29, 34]. For conjugacy-class sizes the criterion has a different group-theoretic conclusion [15 Lemma 2.1].

Such a Sylow subgroup is a direct factor. Our Theorem 1.1 gives a categorical version of (1). Theorem 1.2 bounds the number of connected components formed by the primes that can occur in the prime class graph.

Let $\mathcal{C}$ be a weakly integral braided fusion category over $\mathbb{C}$. Its conjugacy classes are by definition the simple summands $Z_i$ of the canonical algebra $I(\mathbf{1})$ in its Drinfeld center, where $I$ is right adjoint to the forgetful functor $F: Z(\mathcal{C}) \to\mathcal{C}$. Their Frobenius–Perron dimensions are positive integers. We denote by $\Delta_{\mathrm{cl}}(\mathcal{C})$ the graph whose vertices are the primes dividing some $\operatorname{FPdim}(Z_i)$, with an edge between $p$ and $q$ when $pq$ divides a single $\operatorname{FPdim}(Z_i)$. For $\mathcal{C}=\operatorname{Rep}(G)$ the class dimensions are the ordinary class sizes $|g^G|$. As usually, we write $\mathcal{C}_{\mathrm{ad}}$ for the subcategory generated by the objects $X \otimes X^*$ with $X$ simple, and $\mathcal{C}_{\mathrm{pt}}$ for the subcategory of invertible objects and their direct sums.

Here $\mathcal{P}$ is generated by the Sylow $p$-subgroup of $\operatorname{Inv}(\mathcal{C})$, and $\mathcal{B}$ is the sum of the universal grading components whose degrees have order prime to $p$. In particular,

where $\pi(n)$ denotes the prime divisors of $n$.

Under hypothesis (i), one already has $p^{a}\mid\operatorname{FPdim}(\mathcal{C}_{\mathrm{pt}})$ by [5 Theorem 1.2]. The modular pointed splitting and its iteration were given in [5 Proposition 1.3 and Theorem 1.4] in terms of simple dimensions. In the weakly integral modular case, the prime divisors must be taken from the integer squares $\operatorname{FPdim}(X)^2$, as explained in § 3.2. Theorem 1.1 gives the braided splitting for possibly degenerate $\mathcal{C}$, specifies its complement, and characterizes it by the adjoint subcategory. The pointed factor can be degenerate. If $\mathcal{C}$ is nondegenerate, both factors are nondegenerate.

The conjugacy-class prime graph $\Delta_{\mathrm{cl}}(\mathcal{C})$ has vertex set $\bigcup_i\pi(\operatorname{FPdim}(Z_i))$. Two distinct vertices $p$ and $q$ are adjacent if and only if $pq\mid\operatorname{FPdim}(Z_i)$ for some conjugacy class $Z_i$.

We note that this bound is sharp. In $\operatorname{Rep}(D(S_3))$, the nontrivial class dimensions are 4 and 9, so the graph consists of the two isolated vertices 2 and 3 (Example 6.5).

The history of the finite-group bound involves two graphs. In 1981 Kazarin classified groups admitting a pair of isolated conjugacy classes [31 main theorem]. In 1990 Bertram, Herzog and Mann proved that the graph on noncentral conjugacy classes, with adjacency defined by a common prime divisor of their sizes, has at most two components [3]. This also bounds the number of components of the prime graph. Indeed, both are obtained from the incidence graph between primes and noncentral classes by joining vertices on one side with a common neighbor. Alfandary gave the result directly for the graph on primes in 1994 and characterized the disconnected case in terms of a Frobenius quotient [1]. Theorem 1.2 extends the component bound from $\operatorname{Rep}(G)$ to arbitrary weakly integral braided fusion categories.

For comparison, in [36], Natale and Rodríguez studied the graph built from the dimensions of simple objects. For integral nondegenerate braided categories their Theorem 1.1 gives at most three components in general and at most two under solvability. The bound of two also holds when all Tannakian subcategories are pointed [36 Remark 2.2 and Proposition 10.4] and for every group-theoretical modular category [36 Theorem 1.2(ii)]. For a modular category the class dimensions are precisely $\operatorname{FPdim}(X)^2$, $X\in\operatorname{Irr}(\mathcal{C})$ (Lemma 6.2(1)). In the integral modular case, squaring preserves every prime support, so these two graphs coincide. Theorem 1.2 therefore gives the bound of two for the simple-dimension graph of every integral modular category. For strictly weakly integral modular categories the graph here uses the integer squares $\operatorname{FPdim}(X)^2$.

The solvability assertion also has a precursor in the group-theoretical modular case. The proof of [36 Theorem 1.2(ii)] realizes $\mathcal{C}$ as a braided factor of a twisted double $\operatorname{Rep}(D^{\omega}(G))$ with pointed complement, so the two categories have the same prime graph. When that graph is disconnected, $G/Z(G)$ is Frobenius with abelian kernel and complement [36 Theorem 8.5(iii)]. Thus $G$ is solvable. Solvability passes from $\operatorname{Vec}^{G}_{\mathcal{C}}$ to its center and then to $\mathcal{C}$ by [24 Proposition 4.5(i),(ii)]. Theorem 1.2 obtains solvability from disconnectedness for every weakly integral braided fusion category.

Two properties of the class objects enter the graph arguments. First, Proposition 2.2 identifies the underlying class objects $Y_i=F(Z_i)$ as internal endomorphism objects, and Proposition 2.3 proves that their character values are nonnegative. Using orthogonality, Theorem 3.1 gives the regular-product formula

The coefficient of the tensor unit gives the divisibility used in Theorem 1.1. In the modular case this recovers [11 Theorem 1.16]. Second, Theorem 2.6 gives the individual divisibility (23) under restriction to a braided fusion subcategory. Under equivariantization of a non-pointed base, every component of the new graph contains a prime from the base graph (Theorem 7.6). The pointed case follows from the character-fiber formula (169) of Proposition 7.7.

The proof of Theorem 1.2 has two parts. Theorem 6.4 establishes the bound for modular categories. Proposition 6.7 constructs a modular extension with the same class graph for each weakly integral slightly degenerate category. De-equivariantization by the even part of the Müger center reduces the general bound to these two cases (Theorem 7.11). For solvability, Proposition 8.7 uses centralizer descent to reduce a hypothetical minimal counterexample to a weakly group-theoretical category, where the structure of its core gives the contradiction (Theorem 8.9). The stronger assertion that class characters vanish across components holds for categories tensor equivalent to $\operatorname{Rep}(G)$ and for untwisted Drinfeld doubles (Propositions 9.3 and 9.5). In the modular case it also holds when every Tannakian subcategory is pointed, by (107) in the proof of Theorem 6.4. The general vanishing question remains open.

Section 2 proves that character values on class images are nonnegative and that class dimensions divide under restriction. Section 3 derives the Sylow splitting of Theorem 1.1 from the regular-product identity (32) of Theorem 3.1. Section 4 treats products and gradings and proves solvability for prime-power class dimensions (Theorem 4.10). Section 5 uses coprime class dimensions to prove the two-component bound when every Tannakian subcategory is pointed (Theorem 5.16). Section 6 treats modular categories, slightly degenerate categories, and categories with cyclic Tannakian Müger center. In the slightly degenerate case, the two fermion quotients give all class dimensions (Proposition 6.22). Section 7 proves the general two-component bound by equivariantization (Theorem 7.11), and Section 8 deduces solvability from disconnectedness by centralizer descent (Theorem 8.9). Section 9 studies vanishing across components for finite-group representation categories and untwisted Drinfeld doubles. Appendix A uses the relative vanishing theorem (Theorem A.1) to bound the total weight of characters vanishing on a class image (Corollary A.6).

Class objects, formal codegrees and restriction

For a fusion category $\mathcal{C}$, write $\operatorname{Irr}(\mathcal{C})$ for its simple isomorphism classes and $[V:X] = \dim\operatorname{Hom}(X,V)$ for the multiplicity of a simple $X$ in $V$. The pointed subcategory $\mathcal{C}_{\mathrm{pt}}$ is generated by the invertible simple objects, equivalently those with dimension one. Their isomorphism classes form a group $\operatorname{Inv}(\mathcal{C})$, which is abelian when $\mathcal{C}$ is braided. The adjoint subcategory $\mathcal{C}_{\mathrm{ad}}$ is generated by $X \otimes X^*$ for $X \in\operatorname{Irr}(\mathcal{C})$. The universal grading is the faithful grading

For weakly integral $\mathcal{C}$, the subcategory $\mathcal{C}_{\mathrm{ad}}$ is integral [27]. The Deligne tensor product is denoted by $\boxtimes$; its simple objects are external tensor products and its dimension is the product of the dimensions of its factors.

Recall that a fusion category is nilpotent if repeated passage to the adjoint subcategory eventually gives Vec. It is weakly group-theoretical if it is categorically Morita equivalent to a nilpotent fusion category, and solvable if it is Morita equivalent to one obtained from Vec by successive extensions by cyclic groups of prime order. Here categorical Morita equivalence means equivalence to the dual category of an indecomposable semisimple module category; see [24].

All categories are over $\mathbb{C}=\mathcal{C}$. A fusion category $\mathcal{B}$ is weakly integral if $\operatorname{FPdim}(\mathcal{B})$ is an integer, and integral if every simple object has integer Frobenius–Perron dimension. A braided category is nondegenerate if its Müger center is $\operatorname{Vec}$, slightly degenerate if its Müger center is braided equivalent to $\operatorname{sVec}$, and symmetric if its Müger center is the entire category. For weakly integral categories we use the canonical positive spherical structure, under which a nondegenerate braided fusion category is modular. Unless otherwise specified, $\mathcal{C}$ is weakly integral and braided. Write

where $I$ is the right adjoint of the forgetful functor $F$. Since $K_0(\mathcal{C})$ is commutative, the canonical algebra is multiplicity-free [38 Theorem 2.13]. It decomposes as

The $Z_i$ are the conjugacy-class objects. Put

Under weak integrality both $\operatorname{FPdim}(Z_i)$ and $f_i$ are positive integers. Here $f_i$ is the formal codegree of the associated algebra character $\mu_i$ of $K(\mathcal{C})=K_0(\mathcal{C})\otimes_{\mathbb{Z}}\mathbb{C}$. Here a character is a unital algebra homomorphism to $\mathbb{C}$. For a braided fusion category $\mathcal{B}$ and a character $\nu$ of $K(\mathcal{B})$, write $Z^\mathcal{B}_\nu$ for the corresponding conjugacy-class object in $Z(\mathcal{B})$. We use unnormalized $S$-matrices throughout. We denote the set of connected components of a graph by $\pi_0$ and the Müger center by $Z_2(\mathcal{C})$ (also written $\mathcal{C}'$).

For $n\in\mathbb{Z}_{>0}$ let $\pi(n)$ be the set of prime divisors of $n$; thus $\pi(1)=\varnothing$. The graph $\Delta_{\mathrm{cl}}(\mathcal{C})$ has vertex set $\bigcup_i\pi(\operatorname{FPdim}(Z_i))$, with an edge between distinct primes $p,q$ if $pq\mid\operatorname{FPdim}(Z_i)$ for some $i$. A class dimension $\operatorname{FPdim}(Z_i)>1$ is said to belong to a component when its prime support does; its support is a clique and hence lies in a unique component. We count zero components for the empty graph and call a graph disconnected when it has at least two nonempty components. The common-divisor graph on the nontrivial class dimensions has the same number of connected components, but not necessarily the same diameter.

Let $X^+=\iota_+(X)$ and $X^-=\iota_-(X)$ denote the two canonical lifts to the center. Their images satisfy

We write $N^W_{X,Y}=[X\otimes Y:W]$ for the fusion multiplicities. For a full fusion subcategory $\mathcal{B}$ and an object $V$, the notation $V\cap\mathcal{B}$ denotes the sum of the simple constituents of $V$ belonging to $\mathcal{B}$, with multiplicities.

For a commutative fusion ring, let $\tau$ denote the coefficient of the unit. The primitive idempotents, the trace decomposition, and character orthogonality are given by