Augmented quasigroups and character algebras

Abstract

The conjugacy classes of groups and quasigroups form association schemes, in which the relation products are defined by collapsing group or quasigroup multiplications. In previous work, sharp transitivity was used to identify association schemes, such as certain Johnson schemes, which cannot appear as quasigroup schemes. Thus quasigroup schemes only constitute a fragment of the full set of all association schemes. Nevertheless, the current paper shows that every association scheme is in fact obtained by collapsing a quasigroup multiplication. In a second application of a similar technique, character quasigroups are constructed for each finite group, as analogues of the character groups of abelian groups, to encode the multiplicative structure of group characters. As infrastructure for these and related results, three key unifying concepts in compact closed categories are established: augmented comagmas, augmented magmas, and augmented quasigroups, the latter serving to capture such diverse structures as groups and Heyting algebras.