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.
 
 
Highlights
Distance regular graphs have many nice algebraic and combinatorial properties and have been extensively studied
In this paper we focus on the Hamming graph H(n, e), whose vertex set X consists of all words of length n on the alphabet {0, 1, . . . , e − 1} and whose edge set E consists of all unordered pairs of vertices differing in exactly one position
As an application of the linear character approach, we provide a formula for the Norton product on each eigenspace Vi of the Hamming graph H(n, e), and use this formula to study the automorphism group Aut(Vi) of the Norton algebra Vi
Summary
Distance regular graphs have many nice algebraic and combinatorial properties and have been extensively studied. Our method is valid for the bilinear forms graph Hq(d, e), whose vertex set is X = Matd,e(Fq) consisting of all d × e matrices over a finite field Fq and whose edge set E consists of unordered pairs xy of vertices x, y ∈ X with rank(x − y) = 1 This is a distance regular graph of diameter d (assuming d e) and can be viewed as a q-analogue of the Hamming graph H(d, e) [3, §9.5.A].
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