Non-symmetric class $2$ association schemes obtained by doubling of skew-Hadamard matrices are non-schurian

TL;DR

This paper demonstrates that non-symmetric class 2 association schemes derived from doubling skew-Hadamard matrices are inherently non-schurian for matrices of order 8 or higher, revealing limitations in their symmetry properties.

Contribution

It introduces a method to construct non-symmetric class 2 association schemes from skew-Hadamard matrices and proves their non-schurian nature for large orders.

Findings

Constructs non-symmetric class 2 association schemes from skew-Hadamard matrices.

Shows these schemes are non-schurian for orders ≥ 8.

Provides a theoretical proof of non-schurian property for the constructed schemes.

Abstract

We can obtain a non-symmetric class $2$ association scheme by a skew-Hadamard matrix. We begin with a skew-Hadamard matrix of order $n$, construct a skew-Hadamard matrix of order $2n$ by doubling construction, and a non-symmetric class $2$ association scheme of order $2n-1$. We will show that the association scheme obtained in this way never be schurian if $n$ is greater than or equal to $8$.