The boundary of the Gelfand-Tsetlin graph: A new approach

TL;DR

This paper introduces a novel approach to understanding the boundary of the Gelfand-Tsetlin graph by deriving a new explicit formula for counting semi-standard Young tableaux, connecting combinatorics, symmetric functions, and representation theory.

Contribution

It provides a new explicit formula for counting skew-shaped Young tableaux, offering a fresh perspective on the Edrei-Voiculescu theorem through symmetric functions.

Findings

New explicit formula for semi-standard Young tableaux counts

Connection between symmetric functions and Gelfand-Tsetlin boundary

Alternative proof of the Edrei-Voiculescu theorem

Abstract

The Gelfand-Tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups. The boundary of the Gelfand-Tsetlin graph has at least three incarnations --- as a discrete potential theory boundary, as the set of finite indecomposable characters of the infinite-dimensional unitary group, and as the set of doubly infinite totally positive sequences. An old deep result due to Albert Edrei and Dan Voiculescu provides an explicit description of the boundary; it can be realized as a region in an infinite-dimensional coordinate space. The paper contains a novel approach to the Edrei-Voiculescu theorem. It is based on a new explicit formula for the number of semi-standard Young tableaux of a given skew shape (or of Gelfand-Tsetlin schemes of trapezoidal shape). The formula is obtained via the theory of symmetric functions, and new Schur-like symmetric functions play a key role in the derivation.