On higher dimensional cocyclic Hadamard matrices

Abstract

Provided that a cohomological model for G is known, we describe a method for constructing a basis for n-cocycles over G, from which the whole set of n-dimensional n-cocyclic matrices over G may be straightforwardly calculated. Focusing in the case n=2 (which is of special interest, e.g. for looking for cocyclic Hadamard matrices), this method provides a basis for 2-cocycles in such a way that representative 2-cocycles are calculated all at once, so that there is no need to distinguish between inflation and transgression 2-cocycles (as it has traditionally been the case until now). When n>2, this method provides an uniform way of looking for higher dimensional n-cocyclic Hadamard matrices for the first time. We illustrate the method with some examples, for n=2,3. In particular, we give some examples of improper 3-dimensional 3-cocyclic Hadamard matrices.