Motivated by the problems appearing in the theory of experimental designs, Bose and Mesner introduced in 1959 a special class of matrix algebras, known nowadays as Bose-Mesner algebras of symmetric association schemes. In the same year, Shah published the paper where he proposed a more general idea: to replace the standard matrix product with the Jordan product. So, in fact, he introduced the objects called later Jordan schemes by Cameron. While the ideas of Bose and Mesner led to a new direction in algebraic graph theory called later algebraic combinatorics by Bannai and Ito, Shah’s idea was not developed at all. Only in 2004 Shah’s approach was analyzed by Bailey. She observed that the symmetrization of any association scheme is a Jordan scheme, which led to the following question posed by Cameron: “Are there any others?”. In my talk, I’m going to present some constructions of Jordan schemes that provide an affirmative answer to Cameron’s question.

This is joint work with M. Klin and Sven Reichard.

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