Association schemes and hypergroups

ABSTRACT

In this paper, we investigate hypergroups which arise from association schemes in a canonical way; this class of hypergroups is called realizable. We first study basic algebraic properties of realizable hypergroups. Then we prove that two interesting classes of hypergroups (partition hypergroups and linearly ordered hypergroups) are realizable. Along the way, we prove that a certain class of projective geometries is equipped with a canonical association scheme structure which allows us to link three objects; association schemes, hypergroups, and projective geometries (see, Section 1.2 for details).

KEYWORDS:

• Association schemes
• hypergroups
• projective geometry

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 05E30(primary)
• 20N20(secondary)

Acknowledgements

This paper was initiated by the conversation with Paul-Hermann Zieschang. The author thanks to him for explaining how one can attach an association scheme to projective geometry by using flags and also for helpful answers for the author’s questions on association schemes. The author also thanks to Christopher French for very helpful conversations which helped the author greatly to shape the paper as well as various comments on the first draft. Finally, the author thanks to the referee for useful comments.