ABSTRACT
The oldest branch of quasigroup representation theory is the combinatorial character theory. The source of the theory is the diagonal action of the multiplication group on the direct square Q2 of a quasigroup Q, as introduced in Section 2.3. The conjugacy classes of a quasigroup Q are defined in Section 6.1 as the orbits of this action. Section 6.2 introduces the quasigroup class functions, complex-valued functions on Q2 constant on these orbits. If Q is finite, the incidence matrices of the conjugacy classes form a basis for a complex vector space of matrices that is actually a commutative algebra, the centralizer ring of Section 6.3. In Section 6.4, this algebra is identified as the algebra of class functions under convolution, while Section 6.5 gives some other interpretations. As outlined in Section 6.6, the matrices of transition to and from a basis of primitive idempotents for the algebra normalize to yield a character table, which is the usual group character table if Q is a group. Section 6.7 shows how the familiar orthogonality relations of group character theory extend to quasigroups. Section 6.8 treats the common case of rank two quasigroups, in which the only diagonal orbits of the multiplication group are the equality and diversity relations. Section 6.9 introduces numerical invariants of entropy and asymptotic entropy for quasigroups. The entropy is determined by the character table of a quasigroup Q, while the asymptotic entropy is determined by the sequence of character tables of powers of Q.
