Characters of Finite Quasigroups IV: Products and Superschemes

For finite loops (as for finite groups), the character table of a direct product is the tensor product of the character tables of the direct factors. This is no longer true for quasigroups. Although non-ℨ and ℨ-quasigroups may have the same character table, the character table of Q × Q determines whether a finite non-empty quasigroup Q lies in ℨ or not. A combinatorial interpretation of the tensor square of a quasigroup character table is obtained, in terms of superschemes, a higherdimensional extension of the concept of association scheme.