Higher transitive quantum groups: theory and models

TL;DR

This paper explores higher levels of transitivity in quantum permutation groups, examines matrix models for their algebras, and provides new insights into quantum groups derived from Hadamard and Weyl matrices.

Contribution

It introduces the concept of $k$-transitivity for quantum groups at $k extgreater 2$, and studies matrix modeling questions related to these quantum groups.

Findings

Analysis of $k$-transitivity for quantum permutation groups.

Introduction of double and triple flat matrix models.

Results on quantum groups from Hadamard and Weyl matrices.

Abstract

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 $k\geq3$. 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.