A Quantized Analogue of the Markov-Krein Correspondence

TL;DR

This paper introduces a quantized analogue of the Markov-Krein correspondence by analyzing the asymptotic behavior of measures derived from unitary group representations and establishing a bijection between measures and Young diagrams.

Contribution

It develops a new quantized framework linking measures and Young diagrams, extending the classical Markov-Krein correspondence to a novel setting.

Findings

Convergence of measures and Young diagrams as group size grows

Explicit moment generating function relationship between limits

Bijection between bounded measures and continual Young diagrams

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 $\mathbf{m}$, 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 $\mathbf{m}$ 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.