Prime factors of fusion-category dimensions: weak group-theoreticity and decomposition

Introduction

Let $D$ be the Frobenius–Perron dimension of a fusion category over $\mathbb{C}$. The category is weakly integral if $D$ is an integer, and weakly group-theoretical (WGT) if it is Morita equivalent to a nilpotent fusion category. Etingof, Nikshych and Ostrik asked whether every weakly integral fusion category is WGT [32, Section 11, Question 2]. The question is equivalent to its braided version: the Drinfeld center has dimension $D^{2}$, and weak group-theoreticity passes from the center to the original category [32, Proposition 4.1].

A prime is repeated if its exponent in $D$ is at least two. Natale proved that at most two repeated primes suffice for weak group-theoreticity in the nondegenerate braided case, and obtained weak group-theoreticity for $D<1800$ and solvability for odd $D<33075$ [41, Theorems 7.4, 8.1 and 8.2]. Dong extended these results to arbitrary braided categories and obtained weak group-theoreticity for $D<120$ without braiding [22, Theorems 1.1–1.2, p. 556, and Corollaries 3.2–3.3]. A useful way to go beyond this theorem is to bound selected exponents while allowing the others to vary independently. In what follows, displayed primes are distinct and $d$ denotes an arbitrary coprime square-free integer. We call $N$ covered when every weakly integral braided fusion category of dimension dividing $N$ is WGT.

For three repeated primes, Theorem 8.3 gives the unrestricted family $p^{5}q^{a}r^{b}d$ at $p<q<r$. A quartic exponent at the middle prime also suffices (Theorem 7.3). Without any ordering condition on the primes, Theorem 8.1 covers $p^{7}q^{5}r^{4}d$. The five-prime theorem covers one quartic exponent, three cubic exponents and one square in any order, as well as the case of five cubic exponents (Theorems 15.1 and 15.5). These statements include every divisor of the indicated dimension. For an arbitrary number of primes, Theorem 4.17 bounds a prime outside a given simple degree in terms of that degree with its largest prime-power factor removed. Lemma 16.1 uses two principal blocks to omit two designated primes. Proposition 16.2 consequently gives sufficient inequalities independent of both exponents. There is also a finite reduction in the opposite direction. For fixed complementary factor $m$ the principal-block argument bounds both $p$ and its exponent in a minimal table of dimension $p^{a}m$ (Theorem 16.5 and Corollary 16.6).

Write $Q(D)=\prod_{v_p(D)\geq2}p^{v_p(D)}$. The odd condition on $Q(D)$ means precisely $4\nmid D$; the dimension itself can be even. Comparing the family theorems gives the following strict thresholds (Theorem 17.1).

The companion paper proves the braided bound $D<5400^{2}$ directly, using the same principal-block and local Weyl arguments. Its treatment of the remaining dimensions uses short residue and norm calculations that can be checked by hand.

The first open braided core is $29160000=2^{6}3^{6}5^{4}=5400^{2}$. Using the cyclotomic estimates of § 21, § 22–§ 24 treat dimensions $p^{6}q^{4}r^{4}$ for arbitrary distinct primes, and § 25–25 treat the two boundary dimensions. The slightly degenerate branch is excluded throughout that family. At its least member a modular obstruction has rank between 101 and 46774. At dimension 5400 a non-WGT fusion category which is itself simple and perfect has rank between 11 and 462 when its fusion ring is commutative, and between 13 and 490 otherwise. In arbitrary dimension we exclude rank 15 for a nontrivial simple perfect integral modular category, giving rank at least 16. In odd dimension the bound 27 is due to Dong and Palcoux [24, Corollary 1.3]. A non-WGT perfect integral fusion category has rank at least 7, and every nontrivial odd-dimensional perfect integral fusion category has rank at least 15 (Corollary 18.2).

A different application concerns braided decomposition. For composite square-free $L$, Section 26 characterizes the existence of a prime integral WGT modular category of dimension $L^{2}$ by a signed divisibility graph. For example, at $L=5\cdot7\cdot29$ the two possible edges require opposite signs at 29, since $5\mid29+1$ and $7\mid29-1$. Every integral modular category of this dimension therefore splits.

Proof strategy. § 2–§ 7 reduce a divisor-minimal braided obstruction to one of two character tables. In the modular case the category is integral, perfect and fusion-simple. In the slightly degenerate case we use the positive table obtained by identifying the transparent fermion with the unit. On a full principal block, columns with different primary twists are orthogonal in the modular case. At 2 in the positive table, the corresponding assertion uses the squares of the twists. For three repeated primes, orthogonality gives two principal simples whose shared character value has prescribed residues (§ 9–§ 10). For four and five repeated primes, norms and inner products of local Weyl columns constrain the degrees (§ 11–§ 17). § 18 treats additional prime powers. § 19 compares the resulting exponent bounds to obtain Table 1. The even four-prime case uses the finite enumeration in Theorem 9.15. The programs and witnesses for the finite calculations are listed under Code, data and certificates.

Reduction to a simple character table

Use the positive spherical structure of a weakly integral category over $\mathbb{C}$, so $D = \operatorname{FPdim}(\mathcal{C})$ and $d_X = \operatorname{FPdim}(X)$ are its categorical dimensions [30, Propositions 8.23 and 8.24]. Write $\mathcal{C}_{\mathrm{pt}}, \mathcal{C}_{\mathrm{ad}}, \mathcal{C}^{\prime}, U(\mathcal{C})$ for its pointed subcategory, adjoint subcategory, Müger center and universal grading group. We call a category perfect if its only invertible simple is the unit, and fusion-simple if its only fusion subcategories are trivial or the whole category. The corresponding ring conditions concern invertible basis elements and based subrings.

Proof. A nontrivial universal grading reduces the question to its neutral component. A proper nontrivial nondegenerate subcategory splits off with its centralizer. A nontrivial Tannakian $\operatorname{Rep}(G)$ reduces to the neutral de-equivariantization, of dimension $D/(|G||H|)$ for grading support $H \leq G$. In each case the smaller dimensions divide $D$ and WGT passes back by extension, equivariantization or tensor product [30, Propositions 8.15 and 8.20], [32, Proposition 4.1], [26, Theorems 3.13(i) and 4.44, Proposition 4.26], [41, Proposition 4.1]. Thus $\mathcal{C} = \mathcal{C}_{\mathrm{ad}}$ is integral by the dimension grading [36, Theorem 3.10], and its symmetric center is $\mathrm{Vec}$ or $\mathrm{sVec}$. Every proper subcategory is integral and WGT, hence pointed in the absence of Tannakian subcategories [42, Theorem 3.1]. It thus lies in $\mathcal{C}_{\mathrm{pt}}$, which is trivial in the modular case by $(\mathcal{C}_{\mathrm{ad}})^{\prime} = \mathcal{C}_{\mathrm{pt}}$, and is $\mathrm{sVec}$ in the slightly degenerate case by [41, Lemma 7.1]. In the latter case pull back based subrings from the fermion quotient.

We use ordinary modular table for the character table of an integral modular category, and positive reduced table for the table of an integral slightly degenerate category after identifying its transparent fermion with the unit. A minimal table is one of the two categorical tables in Lemma 2.1, with the simplicity and perfectness conditions stated there. In particular, its twists, traces and indicators come from the category. Statements formulated for based rings or symmetric integral character data alone use the algebraic hypotheses specified in those statements.

Put $\Delta= D$ in case (i) and $\Delta= D/2$ in case (ii). In (ii) use the fusion ring $R = K_0(\mathcal{C})/(f-1)$ for the transparent fermion $f$. The twist gives $fX \not\cong X, X^*$. Duality-compatible orbit representatives containing $1$ give structure constants $N_{XY}^{Z} + N_{XY}^{fZ}$ and unit coefficient $\delta_{Y,X^*}$. Both rings have the following symmetric integral character data:

For the modular table these identities are standard [29, Sections 8.13–8.14]. For the reduced table, the normalized columns give all characters by [52, Proposition 3.3] (see [7, Proposition 2.7] for the unitary case). The ratio $\Delta/d_Y^2$ is an algebraic integer by [52, Corollary 3.4], hence an integer here. Write $I_s=\{X:d_X=s\}$, $n_s=|I_s|$ and $\sigma\chi_X=\chi_{\sigma X}$. Complex conjugation acts by duality and commutes with cyclotomic Galois action. Thus $f_{\pi_\sigma X}=\sigma f_X=f_X$, the permutation preserves degrees, and symmetry gives

The characters take algebraic-integer values. The unit-coefficient trace is

Proof. The Hopf-link characters $a_Y(X)=S_{XY}/d_Y$ are ring characters [29, Proposition 8.13.11]. Bézout applied to $a_Y(X)$ and $a_X(Y)$ shows $S_{XY}/\operatorname{lcm}(d_X,d_Y)$ is integral, as in the modular criterion of Dong–Palcoux [24, Lemma 2.7 and the proof of Theorem 1.1]. The regular element $R=\sum_Y d_Y[Y]$ satisfies $[T]R=d_TR$. Accordingly $a_T(R)$ is $D$ when $a_T=d$ and zero otherwise. In the positive structure $a_T=d$ means precisely $T\in\mathcal{C}'$ by the double-braiding criterion [29, Proposition 8.20.5(i)]. Symmetry and $a_Y(X^*)=a_Y(X)$ give