Classification of thin Jordan schemes

TL;DR

The paper addresses the problem of classifying Jordan schemes with maximal rank-to-order ratio, proving the sharp bound $|S|/|\Omega| \le \tfrac{3}{2}$ and characterizing the equality case as thin nonregular schemes. It then shows regular thin Jordan schemes are in one-to-one correspondence with Ring Alternative Moufang loops (RA-loops), and constructs an explicit infinite family of autonomous thin regular Jordan schemes arising from RA-loops. The authors develop a comprehensive framework using rainbow algebras, coherent J-algebras, WL-closure, and algebraic autormorphisms to connect Jordan configurations with loop structures and demonstrate the autonomy of these schemes. The results extend the understanding of autonomous Jordan schemes and provide a complete description of the maximal-ratio case, highlighting the deep link between algebraic combinatorics and nonassociative loop theory with potential implications for coherent configurations and their automorphism groups.

Abstract

Jordan schemes generalize association schemes in a similar way as Jordan algebras generalize the associative ones. It is well-known that association schemes of maximal rank are in one-to-one correspondence with groups (so-called thin schemes). In this paper, we classify Jordan schemes of maximal rank-to-order ratio and show that regular Jordan schemes correspond to a special class of Moufang loops, known as Ring Alternative loops.

Theorems & Definitions (42)

• Proposition 2.1
• Proposition 2.2
• proof
• Example
• Example
• Proposition 3.1
• proof
• Definition 3.2
• Proposition 3.3
• proof