The association scheme on the set of flags of a finite generalized quadrangle

TL;DR

This paper explores the association scheme on flags of finite generalized quadrangles, classifies its fusions, and establishes conditions under which such schemes originate from these geometric structures.

Contribution

It provides a complete classification of fusions of the association scheme on flags and links certain schemes to generalized quadrangles.

Findings

All fusions of the scheme are listed.

Descriptions for schemes of classes 2 and 3 are provided.

Certain association schemes are shown to originate from generalized quadrangles.

Abstract

In this paper, the association scheme defined on the flags of a finite generalized quadrangle is considered. All possible fusions of this scheme are listed, and a full description for those of classes 2 and 3 is given. Furthermore, it is showed that an association scheme with appropriate parameters must arise from the flags of a generalized quadrangle. The same is done for one of its 4-class symmetric fusion.