On Jordan schemes

TL;DR

This paper introduces new infinite series of proper Jordan schemes, expanding the understanding of these algebraic-combinatorial objects and addressing a question about their existence distinct from symmetrizations.

Contribution

The paper presents the first infinite series of proper Jordan schemes and initiates the theoretical development of Jordan schemes as a new class of algebraic-combinatorial objects.

Findings

Several infinite series of proper Jordan schemes are described.

First theoretical developments in the study of Jordan schemes are presented.

Existence of Jordan schemes not arising from symmetrizations is confirmed.

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.