Norton algebras of the Hamming Graphs via linear characters

TL;DR

This paper provides a formula for the Norton product on eigenspaces of Hamming graphs using linear characters, explores their automorphism groups, and analyzes their nonassociativity, extending to various related graphs.

Contribution

It introduces a formula for the Norton product on Hamming graph eigenspaces via linear characters and characterizes the automorphism groups of these algebras.

Findings

Norton product formula for Hamming graphs' eigenspaces

Construction of a large automorphism subgroup

Most Norton products are highly nonassociative

Abstract

The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.