On the realization of a class of $\text{SL}(2,\mathbb{Z})$-representations

TL;DR

This paper investigates specific $ ext{SL}(2, ext{Z})$-representations related to modular fusion categories, establishing conditions on primes and exploring the structure of associated categories and extensions.

Contribution

It characterizes when certain irreducible representations can be realized in modular fusion categories and describes their categorical structures and extensions.

Findings

$q - p = 4$ for realizability of the representation

Existence of a non-trivial $ ext{Z}_2$-extension with specific Frobenius-Perron dimensions

Identification of a braided fusion category as a Drinfeld center of a near-group category

Abstract

Let $p<q$ be odd primes, $\rho_1$ and $\rho_2$ be irreducible representations of $\text{SL}(2,\mathbb{Z}_p)$ and $\text{SL}(2,\mathbb{Z}_q)$ of dimensions $\frac{p+1}{2}$ and $\frac{q+1}{2}$, respectively. We show that if $\rho_1\oplus\rho_2$ can be realized as modular representation associated to a modular fusion category $\mathcal{C}$, then $q-p=4$. Moreover, if $\mathcal{C}$ contains a non-trivial \'{e}tale algebra, then $\mathcal{C}\boxtimes\mathcal{C}(\mathbb{Z}_p,\eta)\cong\mathcal{Z}(\mathcal{A})$ as braided fusion category, where $\mathcal{A}$ is a near-group fusion category of type $(\mathbb{Z}_p,p)$. And we show that there exists a non-trivial $\mathbb{Z}_2$-extension of $\mathcal{A}$ that contains simple objects of Frobenius-Perron dimension $\frac{\sqrt{p}+\sqrt{q}}{2}$.