Characters of finite quasigroups VII: Permutation characters

Each homogeneous space of a quasigroup affords a representation of the Bose- Mesner algebra of the association scheme given by the action of the multiplication group. The homogeneous space is said to be faithful if the corresponding representation of the Bose-Mesner algebra is faithful. In the group case, this definition agrees with the usual concept of faithfulness for transitive permutation representations. A permutation char- acter is associated with each quasigroup permutation representation, and specialises appropriately for groups. However, in the quasigroup case the character of the homoge- neous space determined by a subquasigroup need not be obtained by induction from the trivial character on the subquasigroup. The number of orbits in a quasigroup permu- tation representation is shown to be equal to the multiplicity with which its character includes the trivial character.