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Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups

  • J. P. McCarthyEmail author
Article

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.

Keywords

Random walks Finite quantum groups Representation theory 

Mathematics Subject Classification

46L53 (60J05, 20G42) 

Notes

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsCork Institute of TechnologyCorkIreland

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