Diagram versus bundle equivalence for Zt × Z22 -cocyclic Hadamard matrices

Abstract

One of the most promising structural approaches to resolving the Hadamard Conjecture uses the family of cocyclic matrices over Zt×Z22. Two types of equivalence relations for classifying cocyclic matrices over Zt ×Z22 have been independently found. Any cocyclic matrix equivalent by either of these relations to a Hadamard matrix will also be Hadamard. Bundle equivalence is based on algebraic relations between cocycles over any finite group. Diagram equivalence is based on geometric relations between diagrammatic visualisations of cocyclic matrices over the group Zt ×Z22. Here we reconcile the two. We show the group Bund(t) generated by bundle equivalence operations is isomorphic to a subgroup of index 2 in the group Diag(t) generated by diagram equivalence operations, and that Diag(t) = <Bund(t),T> where T is the geometric translation of matrix transposition