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

Let p<q be odd primes and ρ₁ and ρ₂ be irreducible representations of SL(2,Zₚ) and SL(2,Zq) of dimensions p+1/2 and q+1/2 , respectively. We show that if ρ₁⊕ρ₂ 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(Zₚ,η)(A) as a braided fusion category, where A is a near-group fusion category of type (Zₚ,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+√q}{2} .