Two-valenced association schemes and the Desargues theorem

Abstract

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 $$\{1,p\}$$ { 1 , p } -scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable.

Keywords

• 05E30