Proper Jordan schemes exist. First examples, computer search, patterns of reasoning. An essay

Abstract

A special class of Jordan algebras over a field F of characteristic zero is considered. Such an algebra consists of an r-dimensional subspace of the vector space of all square matrices of a fixed order n over F . It contains the identity matrix, the all-one matrix; it is closed with respect to matrix transposition, Schur-Hadamard (entrywise) multiplication and the Jordan product A * B = 1 2 (AB + BA), where AB is the usual matrix product. The suggested axiomatics (with some natural additional requirements) implies an equivalent reformulation in terms of symmetric binary relations on a vertex set of cardinality n. The appearing graph-theoretical structure is called a Jordan scheme of order n and rank r. A significant source of Jordan schemes stems from the symmetrization of association schemes. Each such structure is called a non-proper Jordan scheme. The question about the existence of proper Jordan schemes was posed a few times by Peter J. Cameron.In the current text an affirmative answer to this question is given. The first small examples presented here have orders n = 15, 24, 40. Infinite classes of proper Jordan schemes of rank 5 and larger are introduced. A prolific construction for schemes of rank 5 and order n =The text is written in the style of an essay. The long exposition relies on initial computer experiments, a large amount of diagrams, and finally is supported by a number of patterns of general theoretical reasonings. The essay contains also a historical survey and an extensive bibliography.