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Isolated Partial Hadamard Matrices and Related Topics

We study the partial Hadamard matrices [Formula: see text] which are isolated, under the assumption that the entries [Formula: see text] are roots of unity, or more generally, under the assumption that the combinatorics of H comes from vanishing sums of roots of unity. We first review the various conjectures on the subject, and then we present several new results, regarding notably the isolation of the master Hadamard matrices, [Formula: see text], and the structure of the isolated matrices arising via the McNulty-Weigert construction. We discuss then the notion of isolation, in some related contexts, of the magic unitary matrices, and of the quantum permutation groups.