On Jordan schemes

Abstract: In 2003 Peter Cameron introduced the concept of a Jordan scheme and asked whether there exist Jordan schemes which are not symmetrisations of coherent configurations (proper Jordan schemes). The question was answered affirmatively by the authors last year and some of the examples were presented in an essay uploaded to the arXiv. In this paper we describe several infinite series of proper Jordan schemes and present first developments in the theory of Jordan schemes - a new class of algebraic-combinatorial objects.