# Cohomology-Developed Matrices -- constructing families of weighing matrices and automorphism actions

## TL;DR

This paper introduces a cohomology-based framework for constructing families of weighing matrices with specific automorphism group actions, generalizing previous methods and producing both new and classical matrix families.

## Contribution

It develops a general theory using low-dimensional group cohomology to construct automorphism actions, leading to structured matrices called Cohomology-Developed matrices, extending cocyclic and group development methods.

## Key findings

- Constructed new families of weighing matrices including Paley Conference and Grassmannian types.
- Established a general cohomology-based theory for automorphism group actions.
- Introduced the concept of quasiproducts as a generalization of Kronecker-products.

## Abstract

The aim of this work is to construct families of weighing matrices via their automorphism group action. This action is determined from the $0,1,2$-cohomology groups of the underlying abstract group. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley Conference, the Projective-Space, the Grassmannian, and the Flag-Variety weighing matrices. We develop a general theory relying on low dimensional group-cohomology for constructing automorphism group actions, and in turn obtain structured matrices that we call \emph{Cohomology-Developed matrices}. This "Cohomology-Development" generalizes the Cocyclic and Group Developments. The Algebraic structure of modules of Cohomology-Developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of \emph{quasiproducts}, which is a generalization of the Kronecker-product.