Construction of the outer automorphism of S6 via a complex Hadamard matrix

TL;DR

This paper presents a novel construction of the outer automorphism of S6 using a complex Hadamard matrix and split quaternions, providing new insights into the symmetry structure of the symmetric group.

Contribution

The paper introduces a new method to construct the outer automorphism of S6 leveraging complex Hadamard matrices and algebraic structures, which was not previously known.

Findings

Constructed the outer automorphism of S6 explicitly.

Connected complex Hadamard matrices with group automorphisms.

Utilized split quaternions in the construction process.

Abstract

We give a new construction of the outer automorphism of the symmetric group on six points. Our construction features a complex Hadamard matrix of order six containing third roots of unity and the algebra of split quaternions over the real numbers.