The cohomological reduction method for computing n-dimensional cocyclic matrices

TL;DR

This paper introduces a cohomological reduction method to efficiently compute n-dimensional cocyclic matrices, including higher dimensions, with applications to cocyclic Hadamard matrices, simplifying previous approaches especially for n=2.

Contribution

It presents a novel, unified approach for constructing bases of n-cocycles from known cohomological models, enabling straightforward calculation of cocyclic matrices for any dimension.

Findings

Provides a basis for 2-cocycles that calculates all representatives simultaneously.

Introduces a uniform method for higher-dimensional cocyclic Hadamard matrices.

Includes examples of 3-dimensional cocyclic Hadamard matrices, including improper ones.

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.