Formal Orthogonal Pairs via Monomial Representations and Cohomology
Abstract
A Formal Orthogonal Pair is a pair (A,B) of symbolic rectangular matrices such that ABT=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.
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