Quasi Clifford algebras and systems of orthogonal designs

Abstract

The representation theory of Clifford algebras has been used to obtain information on the possible orders of amicable pairs of orthogonal designs on given numbers of variables. If, however, the same approach is tried on more complex systems of orthogonal designs, such as product designs and amicable triples, algebras which properly generalize the Clifford algebras are encountered. In this paper a theory of such generalizations is developed and applied to the theory of systems of orthogonal designs, and in particular to the theory of product designs.

Gastineau-Hills’ Quasi-Clifford Algebras and Plug-In Constructions for Hadamard Matrices

“…Humphrey Gastineau-Hills fully developed the theory of quasi-Clifford algebras in his thesis of 1980 [5] and published the key results in a subsequent paper [6]. The paper describes the theory of quasi-Clifford algebras in full generality for fields of characteristic other than 2.…”

“…Gastineau-Hills [6] goes on to investigate the Wedderburn structure of the real SQC algebras by first determining the centre of each algebra, and then determining the irreducible representations.…”

“…The work of Gastineau-Hills on quasi-Clifford algebras [5,6] should be better known. In particular, as at April 2018, his paper on the subject [6] had only four citations other than self-citations, according to Google, [3,12,16,22], and only one of these [16] discusses quasi-Clifford algebras and their representation theory to any depth.…”

“…See also the classifications given by de Launey and Smith [4]. The original application of quasi-Clifford algebras and their representation theory was to systems of orthogonal designs [5,6,22]. The current paper applies quasi-Clifford algebras and their representation theory to the study of some plug-in constructions for Hadamard matrices described by the author in 2014 [12].…”

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The quasi-Clifford algebras as described by Gastineau-Hills in 1980 and1982, should be better known, and have only recently been rediscovered. These algebras and their representation theory provide effective tools to address the following problem arising from a plug-in construction for Hadamard matrices: Given λ, a pattern of amicability / anti-amicability, with λ j,k = λ k,j = ±1, find a set of n monomial {−1, 0, 1} matrices D of minimal order such thatMathematics Subject Classification (2010).

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