酉群的表示环与代数起源的马尔可夫过程
The representation ring of the unitary groups and Markov processes of algebraic origin
AI总结：
该研究引入紧致酉群族的新型表示环，并基于此推导出无穷维酉群对偶对象上马尔可夫过程的无穷小生成元的显式表达式。
摘要：
本文分为两部分。第一部分介绍了紧致酉群族U(1), U(2), …的表示环。这一新对象是一个具有无穷维齐次分量的交换分次代数R。对称函数代数充当有限对称群族的表示环，而该新对象则发挥着类似的作用。第一部分的目的是阐述基本定义，为论文第二部分的构建奠定基础。第二部分研究无穷维酉群U(∞)的对偶对象上的一类马尔可夫过程。这些过程是在与Alexei Borodin的合作工作（J. Funct. Anal. 2012；arXiv:1009.2029）中定义的。本文的主要结果是推导出了它们的无穷小生成元的显式表达式。研究表明，这些生成元由某些具有可数个变量的二阶偏微分算子实现，这些算子最初被定义为R上的算子。
英文摘要：
The paper consists of two parts. The first part introduces the representation ring for the family of compact unitary groups U(1), U(2),.... This novel object is a commutative graded algebra R with infinite-dimensional homogeneous components. It plays the role of the algebra of symmetric functions, which serves as the representation ring for the family of finite symmetric groups. The purpose of the first part is to elaborate on the basic definitions and prepare the ground for the construction of the second part of the paper. The second part deals with a family of Markov processes on the dual object to the infinite-dimensional unitary group U(infinity). These processes were defined in a joint work with Alexei Borodin (J. Funct. Anal. 2012; arXiv:1009.2029). The main result of the present paper consists in the derivation of an explicit expression for their infinitesimal generators. It is shown that the generators are implemented by certain second order partial differential operators with countably many variables, initially defined as operators on R.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
