Gastineau-Hills' quasi-Clifford algebras and plug-in constructions for Hadamard matrices

TL;DR

This paper explores the use of Gastineau-Hills' quasi-Clifford algebras to develop new plug-in constructions for Hadamard matrices, addressing key algebraic questions about matrix patterns and minimal order solutions.

Contribution

It applies the representation theory of quasi-Clifford algebras to construct minimal order monomial matrices satisfying specific amicability patterns for Hadamard matrix construction.

Findings

Provides algebraic tools for Hadamard matrix construction

Identifies minimal order solutions for matrix patterns

Enhances understanding of algebraic structures in combinatorial design

Abstract

The quasi-Clifford algebras, and their Wedderburn structure and representation theory, as described by Gastineau-Hills in 1980 and 1982, should be better known, and have only recently been rediscovered. These algebras and their representation theory provide effective tools to address certain questions relating to plug-in constructions for Hadamard matrices. The key question addressed is: Given $\lambda$, a pattern of amicability / anti-amicability, with $\lambda_{j,k}=\lambda_{k,j}=\pm 1$, find a set of $n$ monomial $\{-1,0,1\}$ matrices $\{D_j\}$ of minimal order such that $$ D_j D_k^T - \lambda_{j,k} D_k D_j^T = 0 \quad (j \neq k). $$