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The Isaacs conjecture for spherical fusion categories

From papers

Let C\mathcal{C} be a spherical fusion category over C\mathbb{C}, and let s≥0s\geq 0 be rational. If AA is its fusion ring, a category is ss-Isaacs when, for every character ρ:A→C\rho:A\to\mathbb{C} and every simple object X∈CX\in\mathcal{C}, the number

λs(ρ,X):=(dim⁡C)s(dim⁡Zρ)1−sρ(X)dim⁡X\lambda_s(\rho,X):=(\dim \mathcal{C})^s(\dim Z_\rho)^{1-s}\frac{\rho(X)}{\dim X}

is an algebraic integer, where ZρZ_\rho is the corresponding central object. Isaacs conjecture. Every spherical fusion category is ss-Isaacs for every ss. This extends the Isaacs property introduced for s=0s=0 and is motivated by the fact that every ribbon fusion category is Isaacs; it is a slight modification of the cited conjecture and remains unresolved in the stated generality.

Progress summary

Partially solved

A few important cases and conditional results are known, but the conjecture for all spherical fusion categories and all parameters remains open.

The conjecture asks whether every spherical fusion category is ss-Isaacs for every rational s≥0s\geq 0. The broader assertion for arbitrary fusion categories at s=0s=0 is false: the extended Haagerup category is not Isaacs.

Known results

  • Every ribbon fusion category is 00-Isaacs.
  • Every spherical fusion category is 11-Isaacs.
  • Liu, Palcoux, and Ren’s conjecture that every spherical pseudo-unitary category is Isaacs was disproved by the extended Haagerup example.
  • A 2025 paper proves further divisibility results under ss-Isaacs and generated-subcategory hypotheses.

July 2025 conditional advances

New integral and divisibility results apply broadly to ribbon categories and give conditional statements for spherical categories, including real-nonnegative hypotheses. They do not prove the full conjecture or provide a counterexample to its stated spherical form.

Current status (as of August 2026): the cases s=1s=1 for spherical categories and s=0s=0 for ribbon categories are settled, while the all-ss spherical conjecture remains open.

Sources
Sources & referencesView supporting material

Primary source

Pavel Etingof, Dmitri Nikshych and Victor Ostrik, “On a necessary condition for unitary categorification of fusion rings”, arXiv:2102.13239 (2023).

algebraic-integersfusion-categories

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