Reconstructing a generalized quadrangle from the Penttila-Williford $4-$class association scheme

TL;DR

This paper characterizes the Penttila-Williford association scheme by identifying specific parameters and clique properties, enabling reconstruction of the underlying generalized quadrangle from the scheme.

Contribution

It provides a characterization theorem that reconstructs a generalized quadrangle from a 4-class association scheme with certain parameters and clique assumptions.

Findings

Association scheme with specified parameters is uniquely the Penttila-Williford scheme.

Reconstruction of the generalized quadrangle is possible from the scheme.

Conditions on maximal cliques are crucial for the characterization.

Abstract

Penttila and Williford constructed a $4-$class association scheme from a generalized quadrangle with a doubly subtended subquadrangle. We show that an association scheme with appropriate parameters and satisfying some assumption about maximal cliques must be the Penttila-Williford scheme.