有限量子群上的随机游动：量子群的Diaconis-Shahshahani理论
Random Walks on Finite Quantum Groups: Diaconis-Shahshahani Theory for Quantum Groups
摘要：
有限群上的随机游走收敛到随机的概念-以及$k$转换后测量到随机的距离的方法-从经典情况推广到有限量子群上的随机游动的情况。研究有限群上遍历随机游动的一个中心工具是Diaconis和Shahshahani的上界引理。上界引理使用群的表示论来生成到随机的距离的上界，因此可以用于确定遍历游动的收敛速度。量子群的表示理论非常容易理解，并且与典型群的表示论非常相似。这允许将上界引理推广为量子群的上界引理。利用量子Diaconis-Shahshahani上界引理研究了典型群$\mathbb{Z}_n$、$\mathbb{Z}_2^n$、对偶群$\widehat{S_n}$以及Kac & Paljutkin和Sekine的“真正”量子群上遍历随机游动的收敛性.
英文摘要：
The concept of a random walk on a finite group converging to random - and a way of measuring the distance to random after $k$ transitions - is generalised from the classical case to the case of random walks on finite quantum groups. 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 the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The representation theory of quantum groups is very well understood and is remarkably similar to the representation theory of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for quantum groups. The Quantum Diaconis--Shahshahani Upper Bound Lemma is used to study the convergence of ergodic random walks on classical groups $\mathbb{Z}_n$, $\mathbb{Z}_2^n$, the dual group $\widehat{S_n}$ as well as the `truly' quantum groups of Kac & Paljutkin, and Sekine.
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