Formal orthogonal pairs via monomial representations and cohomology

A formal orthogonal pair is a pair (A, B) of symbolic rectangular matrices such that AB^T = 0. It can be applied for the construction of Hadamard and weighing matrices. In this paper we introduce a systematic way for constructing such pairs. Our method involves representation theory and group cohomology. The orthogonality property is a consequence of non-vanishing maps between certain cohomology groups. This construction has strong connections to the theory of association schemes and (weighted) coherent configurations. Our techniques are also capable for producing (anti-) amicable pairs. A handful of examples are given.