Equivalences of ${\mathbb Z} _t \times {\mathbb Z}_2^2$-cocyclic Hadamard matrices

TL;DR

This paper explores algebraic and geometric equivalences of cocyclic Hadamard matrices over ${ m Z}_t imes { m Z}_2^2$, revealing a new geometric equivalence corresponding to matrix transposition, aiding classification efforts.

Contribution

It translates algebraic cocycle equivalences into diagrammatic forms and identifies a new geometric equivalence related to matrix transposition.

Findings

Algebraic and geometric equivalences are compatible for classifying cocyclic Hadamard matrices.

A new geometric equivalence corresponding to matrix transposition is identified.

Diagrammatic translation helps in understanding matrix symmetries and classifications.

Abstract

One of the most promising structural approaches to resolving the Hadamard Conjecture uses the family of cocyclic matrices over ${\mathbb Z} _t \times {\mathbb Z}_2^2$. Two types of equivalence relations for classifying cocyclic matrices over ${\mathbb Z} _t \times {\mathbb Z}_2^2$ have been found. Any cocyclic matrix equivalent by either of these relations to a Hadamard matrix will also be Hadamard. One type, based on algebraic relations between cocycles over any finite group, has been known for some time. Recently, and independently, a second type, based on four geometric relations between diagrammatic visualisations of cocyclic matrices over ${\mathbb Z} _t \times {\mathbb Z}_2^2$, has been found. Here we translate the algebraic equivalences to diagrammatic equivalences and show one of the diagrammatic equivalences cannot be obtained this way. This additional equivalence is shown to be the geometric translation of matrix transposition.