On higher dimensional cocyclic Hadamard matrices
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.
(co)homological model cocyclic matrix proper/improper higher dimensional hadamard matrix