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Séminaire Lotharingien de Combinatoire, 84B.68 (2020), 12 pp.

# Assaf Goldberger and Ilias Kotsireas

# Formal Orthogonal Pairs Via Monomial Representations and Cohomology

**Abstract.**
A *formal orthogonal pair* is a pair (*A*,*B*) of symbolic rectangular matrices such that *A**B*^{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.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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