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

Akihide Hanaki,Masayoshi Yoshikawa

doi:10.5802/alco.167

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.

Highlights

From a rank 3 non-symmetric association scheme of order n − 1, we construct a pair of association schemes (D, Q) with the following properties
D and Q have the same character tables but their Frobenius–Schur indicators are different. These properties are similar to the pair of the dihedral group D8 and the quaternion group Q8 of order 8
In the theory of association schemes, the adjacency algebras are mainly considered over the complex number field

Summary

Introduction

From a rank 3 non-symmetric association scheme of order n − 1, we construct a pair of association schemes (D, Q) with the following properties. D and Q have the same character tables but their Frobenius–Schur indicators are different. These properties are similar to the pair of the dihedral group D8 and the quaternion group Q8 of order 8. In the theory of association schemes, the adjacency algebras are mainly considered over the complex number field. It is known that a rank 3 non-symmetric association scheme of order n − 1 exists if and only if there exists a skew-Hadamard matrix of order n with n ≡ 0 (mod 4).

Preliminaries

Construction

The character tables of D and Q

Irreducible representations and rational adjacency algebras

Remark