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A Formal Orthogonal Pair is a pair $(A,B)$ of symbolic rectangular matricessuch that $AB^T=0$. It can be applied for the construction of Hadamard andWeighing matrices. In this paper we introduce a systematic way for constructingsuch pairs. Our method involves Representation Theory and Group Cohomology. Theorthogonality property is a consequence of non-vanishing maps between certaincohomology groups. This construction has strong connections to the theory ofAssociation Schemes and (weighted) Coherent Configurations. Our techniques arealso capable for producing (anti-) amicable pairs. A handful of examples aregiven.

Formal Orthogonal Pairs via Monomial Representations and Cohomology