Interpolated family of non group-like simple integral fusion rings of Lie type

TL;DR

This paper constructs an infinite family of non-group-like simple integral fusion rings of Lie type, interpolating across all integers q≥2, and explores their categorification and structural properties relevant to fusion categories.

Contribution

It introduces a new interpolated family of fusion rings R_q for all q≥2, including non-prime powers, and analyzes their categorification and structural implications.

Findings

Provides the first infinite family of non group-like simple integral fusion rings.

Shows that interpolated fusion rings R_q cannot be braided when q is not a prime power.

Establishes a link between simplicity of Drinfeld centers and non-braided fusion categories.

Abstract

This paper is motivated by the quest of a non-group irreducible finite index depth 2 maximal subfactor. We compute the generic fusion rules of the Grothendieck ring of Rep(PSL(2,q)), q prime-power, by applying a Verlinde-like formula on the generic character table. We then prove that this family of fusion rings R_q interpolates to all integers q>=2, providing (when q is not prime-power) the first example of infinite family of non group-like simple integral fusion rings. Furthermore, they pass all the known criteria of (unitary) categorification. This provides infinitely many serious candidates for solving the famous open problem of whether there exists an integral fusion category which is not weakly group-theoretical. We prove that a complex categorification (if any) of an interpolated fusion ring R_q (with q non prime-power) cannot be braided, and so its Drinfeld center must be simple. In general, this paper proves that a non-pointed simple fusion category is non-braided if and only if its Drinfeld center is simple; and also that every simple integral fusion category is weakly group-theoretical if and only if every simple integral modular fusion category is pointed.