Transversals for association schemes

Abstract

Generalizing a well-known group-theoretical notion we define transversals for (association) schemes. Two results on transversals of schemes are offered. Firstly, we show that a closed subset in a scheme possesses a factor scheme if it possesses a transversal. Secondly, we characterize the Coxeter schemes in terms of transversals. (Coxeter schemes are exactly those schemes which can be identified with the buildings in the sense of Tits). The second result may be viewed as a "thick version" of the characterization of Coxeter groups by the existence of "minimal coset representatives". On the other hand, the characterizing conditions given in this result are similar to the well-known "gate property" defined for chamber systems having a Coxeter matrix as type. Thus, our second main result may be viewed as a unified treatment of these two results.