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.

中文翻译：

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孤立的部分 Hadamard 矩阵和相关主题

我们研究了孤立的部分 Hadamard 矩阵 [Formula: see text]，假设条目 [Formula: see text] 是单位根，或者更一般地说，假设 H 的组合来自于消失的总和团结的根源。我们首先回顾了关于这个主题的各种猜想，然后我们提出了几个新的结果，特别是关于主 Hadamard 矩阵的隔离，[公式：见正文]，以及通过 McNulty-Weigert 构造产生的隔离矩阵的结构。然后，我们在一些相关的上下文中讨论魔法酉矩阵和量子置换群的隔离概念。