Reconstructing a generalized quadrangle with a hemisystem from a $4-$class association scheme

TL;DR

This paper characterizes a specific 4-class association scheme derived from a generalized quadrangle with a hemisystem, using Krein array and triple intersection numbers, advancing understanding of combinatorial structures.

Contribution

It provides a new characterization of the association scheme associated with a generalized quadrangle with a hemisystem via Krein array analysis.

Findings

Characterization of the scheme by Krein array

Use of triple intersection numbers in analysis

Link between association schemes and generalized quadrangles

Abstract

In 2013, van Dam, Martin and Muzychuk constructed a cometric $Q-$ antipodal $4-$class association scheme from a GQ of order $(t^2,t)$, $t$ odd, which have a hemisystem. In this paper we characterize this scheme by its Krein array. The techniques which are used involve the triple intersection numbers introduced by Coolsaet and Juri\v{s}i\'c.