On quantum increasing sequences (again)

Quantum increasing sequences were introduced by S. Curran to characterize free amalgamated independence of an infinite sequence of random variables (over the tail algebra) by means of the so-called quantum spreadability. This is a de Finetti type theorem that in the classical case was first established by C. Ryll-Nardzewski, but the assumptions are formally weaker. The plan of the talk is to discuss the connection of this structure to quantum permutation groups and describe in full generality the quantum subgroups generated by them. This is done by establishing a certain inductive-type framework for generation in quantum groups, a combinatorial description of quantum permutation groups by means of Temperley-Lieb diagrams and recent resolution of Banica's conjecture: there is no intermediate quantum group satisfying $S_5\subset\mathbb{G}\subset S_5^+$.