The association scheme on the set of flags of a finite generalized quadrangle (Q6542058)

The starting point is an association scheme \(\mathcal{X} = (\Omega,\mathcal{R})\), where \(\Omega\) denotes the set of all flags of a given finite generalised quadrangle and \(\mathcal{R}=\{R_0,R_1,\ldots,R_8\}\) is a certain family of binary relations on \(\Omega\). The scheme \(\mathcal{X}\), which is not symmetric by its definition, is shown to be imprimitive and noncommutative. All its intersection numbers are explicitly given. A quotient scheme of \(\mathcal{X}\) turns out to arise from the point-graph of the given generalised quadrangle. Next, it is established that any association scheme with appropriate parameters equals the scheme coming from the set of flags of a finite generalised quadrangle. Furthermore, all possible fusions of the above scheme \(\mathcal{X}\) are listed and a complete description for those of class \(2\) and \(3\) is given. The paper closes with a description and a characterisation of a particular \(4\)-class symmetric fusion of \(\mathcal{X}\).