A geometric approach to orbital recognition in Chevalley-type coherent configurations and association schemes

We say that a coherent configuration is of Chevalley type if its basis relations correspond to orbitals of a geometric action (G,Ω), where G is a finite Chevalley group. If the action is transitive, the resulting configuration is an association scheme. In this paper, we seek to provide an effective universal strategy for orbital recognition in coherent configurations of Chevalley type. This must be preceded by a universal orbital characterization strategy, which we also propose. There are a number of ways to represent the objects in Ω to enable orbital recognition but, invariably, these strategies depend on the structure of the ambient group. On the other hand, a universal strategy does exist, which involves a canonical embedding of Ω in the flag geometry F(G) of G, however this strategy alone does not facilitate fast orbital recognition. In this paper, we further identify these embedded objects in F(G) with certain distinguished elements in the upper Borel subalgebra L of the Lie algebra for G. This reduces orbital recognition to an investigation of systems of equations that arise from vanishing of the Lie product in L . In many instances these systems of equations turn out to be linear. ∗ Both authors were supported by NSF grant DMS–9115473 † Supported by NSF grant DMS–9304580 V. USTIMENKO AND A. WOLDAR/AUSTRALAS. J. COMBIN. 67 (2) (2017), 166–202 167