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.
Highlights
For any binary operation ∗ defined on a set X with indeterminates x0, x1, . . . , xn taking values from X, it is well known that the number of ways to insert parentheses into the expression x0 ∗ x1 ∗ · · · ∗ xn is the ubiquitous Catalan number Cn := 1 n+1 2n n, which enumerates hundreds of other families of objects [23, 24]
In this paper we study the nonassociativity of the so-called Norton algebras, whose construction relies on the notion of distance regular graph, an important topic in algebraic combinatorics [5, 7, 9]
We focus on the Norton product on the eigenspace V1 of the second largest eigenvalue of Γ, where Γ a member of the following four important families of distance regular graphs: the Johnson graphs, Grassmann graphs, Hamming graphs, and dual polar graphs
Summary
For any binary operation ∗ defined on a set X with indeterminates x0, x1, . . . , xn taking values from X, it is well known that the number of ways to insert parentheses into the expression x0 ∗ x1 ∗ · · · ∗ xn is the ubiquitous. We focus on the Norton product on the eigenspace V1 of the second largest eigenvalue of Γ, where Γ a member of the following four important families of distance regular graphs: the Johnson graphs, Grassmann graphs, Hamming graphs, and dual polar graphs. The Norton products in Theorem 1 are either associative or totally nonassociative except for the second case This case is especially interesting as it provides a new interpretation for the sequence A000975 on OEIS [21] with deep algebraic and combinatorial background and is a natural higher-dimensional extension of the double minus operation coming from a somewhat surprising context.
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