Nonassociativity of the Norton Algebras of some distance regular graphs

TL;DR

This paper investigates the nonassociativity of Norton algebras derived from certain distance regular graphs, providing quantitative measures and revealing cases of maximal nonassociativity with interesting mathematical connections.

Contribution

It offers a precise measurement of nonassociativity for Norton products on eigenspaces of specific graphs, extending previous formulas and identifying exceptional cases.

Findings

Norton product is maximally nonassociative in most cases

Identifies two special cases with different nonassociativity behavior

Connects one case to OEIS sequence A000975 and recent mathematical operations

Abstract

A Norton algebra is an eigenspace of a distance regular graph endowed with a commutative nonassociative product called the Norton product, which is defined as the projection of the entrywise product onto this eigenspace. The Norton algebras are useful in finite group theory as they have interesting automorphism groups. We provide a precise quantitative measurement for the nonassociativity of the Norton product on the eigenspace of the second largest eigenvalue of the Johnson graphs, Grassman graphs, Hamming graphs, and dual polar graphs, based on the formulas for this product established in previous work of Levstein, Maldonado and Penazzi. Our result shows that this product is as nonassociative as possible except for two cases, one being the trivial vanishing case while the other having connections with the integer sequence A000975 on OEIS and the so-called double minus operation studied recently by Huang, Mickey, and Xu.