Nonexistence of modular fusion algebras whose kernels are certain noncongruence subgroups

Abstract

A well-known conjecture states that the kernel of representation associated to a modular fusion algebra is always a congruence subgroup. Assuming this conjecture, Eholzer studied modular fusion algebras such that the kernel of representation associated to each of them is a congruence subgroup using the fact that all irreducible representaions of SL(2,Z/pλZ) are classified. He classified all strongly modular fusion algebras of dimension two, three, four and the nondegenerate ones with dimension ≤24. In this paper, we try to imitate Eholzer's work. We classify modular fusion algebras such that the kernel of representation associated to each of them is a noncongruence normal subgroup of Γ:=PSL(2,Z) containing an element (1601). Among such normal subgroups, there exist infinitely many noncongruence subgroups. In a sense, they are the classes of near congruence subgroups. For such a normal subgroup G, we shall show that any irreducible representation of degree not equal to 1 of Γ/G is not associated to a modular fusion algebra.