A Quantized Analogue of the Markov–Krein Correspondence

Abstract

Abstract We study a family of measures originating from the signatures of the irreducible components of representations of the unitary group, as the size of the group goes to infinity. Given a random signature $\lambda $ of length $N$ with counting measure $\textbf {m}$, we obtain a random signature $\mu $ of length $N-1$ through projection onto a unitary group of lower dimension. The signature $\mu $ interlaces with the signature $\lambda $, and we record the data of $\mu ,\lambda $ in a random rectangular Young diagram $w$. We show that under a certain set of conditions on $\lambda $, both $\textbf {m}$ and $w$ converge as $N\to \infty $. We provide an explicit moment-generating function relationship between the limiting objects. We further show that the moment-generating function relationship induces a bijection between bounded measures and certain continual Young diagrams, which can be viewed as a quantized analogue of the Markov–Krein correspondence.