The Boundary of the Gelfand-Tsetlin Graph: New Proof of Borodin-Olshanski's Formula, and its q-analogue

TL;DR

This paper offers a simpler proof of Borodin and Olshanski's formula for the boundary of the Gelfand-Tsetlin graph, extending it to a q-analogue involving skew Schur polynomials and q-specializations.

Contribution

It provides a more direct derivation of a determinantal formula and introduces a new q-generalization related to the q-Gelfand-Tsetlin graph.

Findings

Simplified derivation of the determinantal formula using Cauchy-Binet summation.

New explicit determinantal formula for q-specializations of skew Schur polynomials.

Connection to q-Gelfand-Tsetlin graph and q-Toeplitz matrices.

Abstract

In the recent paper [arXiv:1109.1412], Borodin and Olshanski have presented a novel proof of the celebrated Edrei-Voiculescu theorem which describes the boundary of the Gelfand-Tsetlin graph as a region in an infinite-dimensional coordinate space. This graph encodes branching of irreducible characters of finite-dimensional unitary groups. Points of the boundary of the Gelfand-Tsetlin graph can be identified with finite indecomposable (= extreme) characters of the infinite-dimensional unitary group. An equivalent description identifies the boundary with the set of doubly infinite totally nonnegative sequences. A principal ingredient of Borodin-Olshanski's proof is a new explicit determinantal formula for the number of semi-standard Young tableaux of a given skew shape (or of Gelfand-Tsetlin schemes of trapezoidal shape). We present a simpler and more direct derivation of that formula using the Cauchy-Binet summation involving the inverse Vandermonde matrix. We also obtain a q-generalization of that formula, namely, a new explicit determinantal formula for arbitrary q-specializations of skew Schur polynomials. Its particular case is related to the q-Gelfand-Tsetlin graph and q-Toeplitz matrices introduced and studied by Gorin [arXiv:1011.1769].