Geodesic
Characters of finite quasigroups VII: permutation characters
Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 2, pp. 265-273.

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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 character is associated with each quasigroup permutation representation, and specialises appropriately for groups. However, in the quasigroup case the character of the homogeneous 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 permutation representation is shown to be equal to the multiplicity with which its character includes the trivial character.
Classification : 05E30, 20C99, 20N05
Keywords: quasigroup; association scheme; permutation character 
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Johnson, K. W.; Smith, J. D. H. Characters of finite quasigroups VII: permutation characters. Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 2, pp. 265-273. http://geodesic.mathdoc.fr/item/CMUC_2004__45_2_a6/