Higher transitive quantum groups: theory and models

We investigate the notion of $k$-transitivity for the quantum permutation groups $G\subset S_N^{+}$, with a brief review of the known $k=1,2$ results, and with a study of what happens at $k\ge 3$. We discuss then matrix modelling questions for the algebras $C(G)$, notably by introducing the related notions of double and triple flat matrix model. At the level of the examples, our main results concern the quantum groups coming from the complex Hadamard matrices, and from the Weyl matrices.