论文标题
双层马尔可夫连锁店
Double Coset Markov Chains
论文作者
论文摘要
令$ g $为有限的组。令$ h，k $为$ g $的亚组，$ h \ backslash g / k $ the double coset空间。令$ q $是$ g $的概率，在$ g $上是共轭类（$ q（s^{ - 1} t s）= q（t）$）的常数。 $ g $项目的$ Q $在$ h \ backslash g /k $上驱动的随机步行。这允许使用$ g $的表示理论分析集团链。例子包括凝血碎裂过程和应急表上的天然马尔可夫链。我们的主要示例预测，随机传输在$ gl_n（q）$上通过$ s_n $通过bruhat分解到马尔可夫链上。 $ s_n $上的链条具有锤子固定分布和有趣的混合时间行为。该投影阐明了高斯消除的组合。在此过程中，我们给出了双圈的Hecke代数中的转向总和。讨论了一些与$ g $ $ g $ compact组的双层马尔可夫连锁店的扩展和示例。
Let $G$ be a finite group. Let $H, K$ be subgroups of $G$ and $H \backslash G / K$ the double coset space. Let $Q$ be a probability on $G$ which is constant on conjugacy classes ($Q(s^{-1} t s) = Q(t)$). The random walk driven by $Q$ on $G$ projects to a Markov chain on $H \backslash G /K$. This allows analysis of the lumped chain using the representation theory of $G$. Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on $GL_n(q)$ onto a Markov chain on $S_n$ via the Bruhat decomposition. The chain on $S_n$ has a Mallows stationary distribution and interesting mixing time behavior. The projection illuminates the combinatorics of Gaussian elimination. Along the way, we give a representation of the sum of transvections in the Hecke algebra of double cosets. Some extensions and examples of double coset Markov chains with $G$ a compact group are discussed.
