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 $SL(2,Z/p^{\lambda }Z)$ are classified. He classified all strongly modular fusion algebras of dimension two, three, four and the nondegenerate ones with dimension $\le 24$. 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 $\Gamma :=PSL(2,Z)$ containing an element $\left(\begin{array}{cc}1& 6\\ 0& 1\end{array}\right)$. 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 $\Gamma /G$ is not associated to a modular fusion algebra.