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arXiv · 1811.12218

Two-valenced association schemes and the Desargues theorem

Mitsugu Hirasaka·Kijung Kim·Ilia Ponomarenko
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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.

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Mitsugu Hirasaka, Kijung Kim, Ilia Ponomarenko. 2018-11-29. Two-valenced association schemes and the Desargues theorem. https://arxiv.org/abs/1811.12218

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