The representation ring of the unitary groups and Markov processes of algebraic origin (Q304104)

Jointly with \textit{A. Borodin} the author introduced a family of Markov processes on the dual object of the infinite-dimensional unitary group \(U(\infty)\), [J. Funct. Anal. 263, No. 1, 248--303 (2012; Zbl 1260.60149)]. The main purpose of the present paper is the derivation of explicit expressions for the infinitesimal generators of these processes. In preparation, the author introduces and studies the representation ring \(\mathbb{R}\) for the family of compact unitary groups \(U(1),U(2),\dots\). It is a commutative graded algebra with infinite-dimensional homogeneous components, playing a similar role as the algebra of symmetric functions does for the family of finite symmetric groups.NEWLINENEWLINE In the second half of the paper it is then shown that the envisaged infinitesimal generators are implemented by the second-order partial differential operators with countably many variables \(\{\varphi_n:n\in\mathbb{Z}\}\) and complex parameters \(x\), \(x'\), \(w\), \(w'\), NEWLINE\[NEWLINE\mathbb{D}_{z,z',w,w'}= \sum_{nn}{\partial 2\over\partial\varphi^2_n}+ 2 \sum _{\substack{ n_1,n_2\in\mathbb{Z}\\ n_1>n_2}} A_{1n_2}{\partial^2\over \partial\varphi_{n_1} \partial\varphi_{n_2}}+ \sum_{n\in\mathbb{Z}} B_n{\partial\over \partial\varphi_n},NEWLINE\]NEWLINE where for any indices \(n_1\geq n_2\), NEWLINE\[NEWLINE\begin{multlined} A_{n_1n_2}= \sum^\infty_{p=0} (n_1-n_2+ 2p+ 1)(\varphi_{n_1+p+1}\varphi_{n_2-p}+ \varphi_{n_1+p} \varphi_{n_2-p-1})\\ -(n_1= n_2) \varphi_{n_1}\varphi_{n_2}-2\sum^\infty_{p=1} (n_1-n_2+ 2p) \varphi_{n_1+jp} \varphi_{n^2-p}\end{multlined}NEWLINE\]NEWLINE and for any \(n\in\mathbb{Z}\), NEWLINE\[NEWLINE\begin{multlined} B_n= (n+ w+ 1)(n+ w'+ 1)\varphi_{n+1}+ (n-z-1)(n-z'-1)\varphi_{n-1}\\ -((n-z)(n-z')+ (n+ w)(n+ w'))\varphi_n.\end{multlined}NEWLINE\]NEWLINE There operators are correctly defined on \(\mathbb{R}\).