Characters of Finite Quasigroups

This paper sets out to sketch some of the basics of a character theory for finite non-empty quasigroups, generalising the traditional ordinary character theory for groups. At first sight such a task might seem to have little chance of success, since the group characters are traces of matrix representations which depend essentially on the presence of associativity. However, there are combinatorial theories available providing descriptions of group characters which do not involve matrix representations (e.g. the 'association schemes' of Delsarte [8], the 'coherent configurations' of D. G. Higman [12], and the 'Schur rings' of Tamaschke [20]). Section 2 below presents the relevant notions from quasigroup theory-the multiplication group, quasigroup conjugacy classes, and the space of class functions-which enable one to apply these theories to quasigroups. The absence of an identity element in a quasigroup (more precisely, the absence of a pointed idempotent) forces one to alter one's approach on generalising from groups to quasigroups. Thus one must consider congruences rather than normal subgroups. Further, it is the direct square of the quasigroup that is partitioned by the conjugacy classes, and where the arguments of the characters are taken. With this preparatory work done, the third section works through the application of the combinatorial theories. Their applicability is demonstrated in Theorem 3.1, using an earlier result of quasigroup theory, and then the details of the combinatorics are recapitulated in a form appropriate to this application. Particular interest attaches to observing directly the effect of relaxing the requirement of associativity. Quasigroup characters as class functions and the character table of a quasigroup are given explicitly, and the characters are seen to form an orthonormal basis for the space of class functions. In Theorem 3.6 it is demonstrated that the character table of a quasigroup includes enough information to specify its congruence lattice (which is analogous to the lattice of normal subgroups of a group). (One of the referees has pointed out that Theorem 3.6 may be proved in a more general setting. Since the proof would be on lines similar to ours, we restrict ourselves to a statement of the general version and to pointing out an interesting consequence which appears to be new.) The paper concludes with a straightforward illustrative example.