Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups

Abstract

A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses Fourier analysis on the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The Fourier analysis of quantum groups is remarkably similar to that of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for finite quantum groups. The Upper Bound Lemma is used to study the convergence of ergodic random walks on the dual group \(\widehat{S_n}\) as well as on the truly quantum groups of Sekine.

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Acknowledgements

I would like to thank Adam Skalski; Example 4.2 was developed during a visit to Adam at the Institute of Mathematics of the Polish Academy of Sciences (IMPAN), Warsaw, Poland. This trip was financially supported by IMPAN and also Cork Institute of Technology. I would like to thank Uwe Franz for assisting with Proposition 2.1. I would like to thank Amaury Freslon for encouragement and helpful comments; in particular for help in greatly improving the presentation of the bounds for the random walk on \(\widehat{S_n}\). The rest of the paper was developed during the author’s Ph.D. study at University College Cork, under the supervision of Stephen Wills.

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Communicated by Hans G. Feichtinger.

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McCarthy, J.P. Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups. J Fourier Anal Appl 25, 2463–2491 (2019). https://doi.org/10.1007/s00041-019-09670-4

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Keywords

  • Random walks
  • Finite quantum groups
  • Representation theory

Mathematics Subject Classification

  • 46L53 (60J05, 20G42)