The Young bouquet and its boundary

TL;DR

This paper introduces the Young bouquet, a new combinatorial object that unifies the boundaries of the Young and Gelfand-Tsetlin graphs, providing insights into the representation theory of infinite groups.

Contribution

It defines the Young bouquet and computes its boundary, revealing a cone structure over the Young graph boundary and a degeneration of the Gelfand-Tsetlin boundary, offering a novel explanation for observed phenomena.

Findings

The boundary of the Young bouquet is a cone over the Young graph boundary.

The Young bouquet boundary degenerates to the Gelfand-Tsetlin boundary.

Application to constructing infinite-dimensional Markov processes with determinantal correlation functions.

Abstract

The classification results for the extreme characters of two basic "big" groups, the infinite symmetric group S(infinity) and the infinite-dimensional unitary group U(infinity), are remarkably similar. It does not seem to be possible to explain this phenomenon using a suitable extension of the Schur-Weyl duality to infinite dimension. We suggest an explanation of a different nature that does not have analogs in the classical representation theory. We start from the combinatorial/probabilistic approach to characters of "big" groups initiated by Vershik and Kerov. In this approach, the space of extreme characters is viewed as a boundary of a certain infinite graph. In the cases of S(infinity) and U(infinity), those are the Young graph and the Gelfand-Tsetlin graph, respectively. We introduce a new related object that we call the Young bouquet. It is a poset with continuous grading whose boundary we define and compute. We show that this boundary is a cone over the boundary of the Young graph, and at the same time it is also a degeneration of the boundary of the Gelfand-Tsetlin graph. The Young bouquet has an application to constructing infinite-dimensional Markov processes with determinantal correlation functions.