A construction of pairs of non-commutative rank 8 association schemes from non-symmetric rank 3 association schemes

TL;DR

This paper constructs a pair of non-commutative rank 8 association schemes from a non-symmetric rank 3 scheme, revealing new structural insights and similarities to classical groups.

Contribution

It introduces a novel construction of non-commutative association schemes with identical character tables but different Frobenius-Schur indicators.

Findings

The constructed schemes have the same character table.

They exhibit different Frobenius-Schur indicators.

Structures of their adjacency algebras are determined over the rationals.

Abstract

We construct a pair of non-commutative rank 8 association schemes from a rank 3 non-symmetric association scheme. For the pair, two association schemes have the same character table but different Frobenius-Schur indicators. This situation is similar to the pair of the dihedral group and the quaternion group of order 8. We also determine the structures of adjacency algebras of them over the rational number field.