A Concrete Model for the Quantum Permutation Group on 4 Points

Abstract

In 2019, Jung–Weber gave an example of a concrete magic unitary M, which defines a $C^{*}$-algebraic model of the quantum permutation group $S_4^{+}$. We show with the help of a computer that there exist no polynomials up to degree 50 separating the entries of M from the generators of $C(S_4^{+})$. This indicates that the magic unitary M might already define a faithful model of $S_4^{+}$.

Keywords:

• quantum permutation group
• magic unitary
• compact quantum group

Acknowledgments

This work has been part of the first author’s Bachelor’s thesis.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 An implementation of our algorithms can be found at https://github.com/nfaross/model-s4plus.

2 An implementation of our algorithms can be found at https://github.com/nfaross/model-s4plus.

3 An implementation of our algorithms can be found at https://github.com/nfaross/model-s4plus.

Additional information

Funding

The second author has been supported by the SFB-TRR 195 (this work is a contribution to the SFB-TRR 195), the Heisenberg program of the DFG and OPUS-LAP Quantum groups, graphs and symmetries via representation theory.