Abstract
The main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition, the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique, we prove a new result: given a prime p, any {1,p}-scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable.
Highlights
An association scheme can be thought as a partition of the arcs of a complete directed graph into digraphs connected via special regularity conditions
Numerous examples of associative schemes include the orbital schemes of transitive permutation groups, the Cayley schemes corresponding to Schur rings, the schemes of distance-regular graphs, etc., see [3]
It is widely believed that the theory of association schemes is one of the most important branches of algebraic combinatorics
Summary
An association scheme can be thought as a partition of the arcs of a complete directed graph into digraphs connected via special regularity conditions. The schurian and separable quasi-thin schemes were characterized in [12] This result can be deduced from an analog of Theorem 1.1, in which the Desarguesian condition is replaced by a weaker one (namely, the amount of the required Desarguesian configuration is reduced). There are many non-schurian and non-separable meta-thin {1, p}-schemes for which that group is elementary abelian of order p2 [8]. Theorem 1.4 Given a prime p, any {1, p}-scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable. Except for the above-mentioned examples from [8], not so much is known on the schurity and separability of a {1, p}-scheme with thin residue isomorphic to an elementary abelian p-group of rank two. An elementary abelian p-group of order pm is denoted by E pm
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