Two-valenced association schemes and the Desargues theorem

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 e...

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Bibliographic Details
Published inarXiv.org
Main Authors Hirasaka, Mitsugu, Kim, Kijung, Ponomarenko, Ilia
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 29.11.2018
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Summary: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.
ISSN:2331-8422