Two constructions of Markov chains on the dual of U(n)

TL;DR

This paper introduces two novel methods for constructing Markov chains related to the representation theory of the infinite-dimensional unitary group, connecting combinatorics, quantum walks, and representation restrictions.

Contribution

It presents two new constructions of Markov chains on the dual of U(n), linking combinatorial and quantum approaches to representation theory.

Findings

First construction uses Littlewood-Richardson coefficients

Second construction involves quantum random walks on group von Neumann algebra

Both methods elucidate the structure of Markov chains on U(n)

Abstract

We provide two new constructions of Markov chains which had previously arisen from the representation theory of the infinite-dimensional unitary group. The first construction uses the combinatorial rule for the Littlewood-Richardson coefficients, which arise from tensor products of irreducible representations of the unitary group. The second arises from a quantum random walk on the group von Neumann algebra of U(n), which is then restricted to the center. Additionally, the restriction to a maximal torus can be expressed in terms of weight multiplicities, explaining the presence of tensor products.