Structure constants, Isaacs property and Extended Haagerup fusion categories

TL;DR

This paper introduces an abstract Isaacs property for fusion rings, explores its categorical analogue, and demonstrates that Extended Haagerup fusion categories do not satisfy this property, impacting the understanding of their structure.

Contribution

It defines an abstract Isaacs property for fusion rings, compares it with existing properties, and applies it to show Extended Haagerup categories lack this property.

Findings

Extended Haagerup fusion categories do not satisfy the Isaacs property.

The Isaacs property is positioned between integrality and 1-Frobenius properties.

The result refutes previous conjectures and clarifies the structure of EH1.

Abstract

This paper presents an abstract Isaacs property that involves the Fourier transform for fusion rings, which may be non-commutative, thus expanding upon the commutative version described in [12]. A categorical version of this property was subsequently introduced in [8] for any spherical fusion category, matching with our abstract version in the pseudo-unitary case. We demonstrate that the Isaacs property occupies a distinct position, falling between the integrality of structure constants and the 1-Frobenius properties, in the commutative case. We show that the Extended Haagerup fusion categories, denoted as EHi, do not satisfy the Isaacs property. This finding provides a negative response to [8, Question 5.8], refutes [12, Conjecture 2.5], and recovers that EH1 lacks a braiding structure.