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

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 $$ D_j D_k^T - \lambda_{j,k} D_k D_j^T = 0 \quad (j \neq k). $$