Nonexistence of modular fusion algebras whose kernels are certain noncongruence subgroups

概要

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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, \bm{Z}/p<SUP>λ</SUP>\bm{Z}) 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 Eholzers 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, \bm{Z}) containing an element \left(\begin{array}{ll} 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 Γ/G is not associated to a modular fusion algebra.
社団法人日本数学会の論文
2003-07-01