On the realization of a class of SL (2, Z) representations

Let p<q be odd primes and ₁ and ₂ be irreducible representations of SL (2, Z₏) and SL (2, Zₐ) of dimensions p+12 and q+12, respectively. We show that if ₁₂ can be realized as a modular representation associated with a modular fusion category C, then q-p=4. Moreover, if C contains a non-trivial étale algebra, then C (Z₏, ) (A) as a braided fusion category, where A is a near-group fusion category of type (Z₏, p), which gives a partial answer to the conjecture of D. Evans and T. Gannon. We also show that there exists a non-trivial Z₂ -extension of A that contains simple objects of Frobenius–Perron dimension p+q2.