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

Primarily AI-generated textHuman understanding: some partsmath.QA — Quantum Algebramath.CT — Category Theorymath.GR — Group Theorymath.MP — Mathematical Physicsmath.RA — Rings and Algebrasmath.RT — Representation Theory

Contributed by Sebastian Burciu ↗, Jingcheng Dong ↗, Sebastien Palcoux ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

Let DD be the Frobenius–Perron dimension of a weakly integral fusion category and Q(D)Q(D) its square-full part. For primes p<q<r<sp<q<r<s and nonnegative integers a,ba,b, we prove that a braided category of dimension DD is weakly group-theoretical whenever Q(D)∣p5qarbQ(D)\mid p^{5}q^{a}r^{b} or Q(D)∣p3q3rasbQ(D)\mid p^{3}q^{3}r^{a}s^{b}. Without a braiding assumption, D∣pqrasbD\mid pqr^{a}s^{b} implies solvability. We also give criteria for adjoining a prime power of arbitrary exponent. For fixed mm, the braided problem for dimensions dividing ℓem\ell^{e}m, with ℓ∤m\ell\nmid m prime and e≥0e\ge 0, reduces to finitely many dimensions. A braided category of dimension DD is weakly group-theoretical if Q(D)<29160000Q(D)<29160000, and solvable if Q(D)Q(D) is odd and Q(D)<6087156075Q(D)<6087156075. Without a braiding assumption, the corresponding conclusions hold for D<5400D<5400 and for odd D<121275D<121275, respectively. We also obtain rank bounds and criteria for decomposing integral modular categories into braided factors.

Provenance statement

GPT-6 Astra served as a research assistant in developing and drafting the arguments, identifying references, and checking finite calculations. The authors are continuing their review and have yet to reach a complete understanding of all the proofs.

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Version history

  1. v1 · 229 pagesSubmitted by Sebastien PalcouxRevised after moderator-requested changesCurrentNov 10, 2026