(Joint with R. Ben-Av, X. Chen, G. Dula, K. Okoudjou) Some recent papers discuss a method of generation of complex line configurations from association schemes. If the scheme is Schurian, subordinate to a group G, then G is an automorphism subgroup of the configuration. However, G then acts on the points, without re-phasing. In this work we propose an extension to weighted Schemes, which introduces G actions that involve re-phasing. Our extension may also accommodate algebraic configurations, in which G is also allowed to act by Galois transformations. This construction involves the study of the cohomology groups of G and of certain subgroups, and is closely related to arithmetic invariants, like the Brauer group.