# Isolated partial Hadamard matrices, and related topics

## TL;DR

This paper investigates isolated partial Hadamard matrices with roots of unity, reviews conjectures, and presents new results on master matrices and related quantum structures, exploring their combinatorial and algebraic properties.

## Contribution

It introduces new findings on the structure of isolated partial Hadamard matrices, especially regarding master matrices and the McNulty-Weigert construction, and discusses their relation to quantum groups.

## Key findings

- New results on master Hadamard matrices
- Analysis of McNulty-Weigert construction
- Conjectures on quantum permutation groups

## Abstract

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.