  We study the isolated partial Hadamard matrices, under the assumption that
the entries are roots of unity, or more generally, under the assumption that
the combinatorics 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 master Hadamard matrices, and 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, with a number of conjectures on the subject.
