Two-valenced association schemes and the Desargues theorem
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Arabian Journal of Mathematics
Mitsugu Hirasaka · Kijung Kim · Ilia Ponomarenko
Two-valenced association schemes and the Desargues theorem
Received: 21 October 2019 / Accepted: 31 October 2019 © The Author(s) 2019
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. Mathematics Subject Classification
05E30
1 Introduction 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 now widely believed that the theory of association schemes is one of the most important branches of algebraic combinatorics. One of the fundamental problems in theory of association schemes is to determine whether a given scheme is schurian, i.e., comes from a permutation group, and/or separable, i.e., uniquely determined by its intersection number array (for the exact definitions, see Sect. 2). In the last 2 decades, these two problems are intensively studied for the two-valenced schemes, see, e.g., [2,6,11,13]; here, an association scheme is said to be twovalenced if the valencies of its basic relations take exactly two values, and if they are 1 and k, the term {1, k}-scheme is also used. An analysis of the known proofs that certain two-valenced schemes are schurian or separable shows that in all cases, the following two properties are significant. The first one is that there are sufficiently many intersection numbers of the scheme in question that are equal to 1; to define this property precisely, we introduce, in Sect. 3, M. Hirasaka · K. Kim Department of Mathematics, Pusan National University, Jang-jeon dong, Busan, Republic of Korea E-mail: [email protected] K. Kim E-mail: [email protected] I. Ponomarenko (B) St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia E-mail: [email protected]
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the saturation condition (a special case of it appeared in [2]). The second property expresses the fact that in a noncommutative “affine” geometry determined by the two-valenced scheme, there are sufficiently many Desarguesian configurations, see Sect. 4. A two-valenced scheme having the fi
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