Abstract
Using a suitably noncommutative flat matrix model, it is shown that the quantum permutation group has free orbitals: that is, a monomial in the generators of the algebra of functions can be zero for trivial reasons only. It is shown that any strictly intermediate quantum subgroup between the classical and quantum permutation groups must have free three-orbitals, and this is used to derive some elementary bounds for the Haar state on degree four monomials in such quantum permutation groups. Exploiting the main character, explicit formulae are given in the case of the quantum permutation group.
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Notes
In the notation of [9, (1)], \(x\in M_0\).
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Acknowledgements
Some of this work goes back to discussions with Teo Banica, and free orbitals in \(S_4^+\) are due to Teo. Thanks to David Roberson to pointing to Definition 4.1 in his pre-print Quantum symmetry vs nonlocal symmetry with Simon Schmidt. The idea for the magic basis which gives the flat matrix model comes from this definition. The question of inner faithfulness was posed by Uwe Franz.
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McCarthy, J. Tracing the orbitals of the quantum permutation group. Arch. Math. 121, 211–224 (2023). https://doi.org/10.1007/s00013-023-01883-w
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DOI: https://doi.org/10.1007/s00013-023-01883-w