# Moscow Mathematical Journal

Volume 14, Issue 1, January–March 2014 pp. 121–160.

The Boundary of the Gelfand–Tsetlin Graph: New Proof of Borodin–Olshanski’s Formula, and its*q*-Analogue

**Authors**: Leonid Petrov

**Author institution:**Department of Mathematics, Northeastern University, 360 Huntington ave., Boston, MA 02115, USA and

Dobrushin Mathematics Laboratory, Kharkevich Institute for Information Transmission Problems, Moscow, Russia

**Summary:**

In a recent paper, 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 infinitedimensional 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.

2010 Mathematics Subject Classification. 05E10, 22E66, 31C35, 46L65.

**Keywords:**Gelfand–Tsetlin graph, trapezoidal Gelfand–Tsetlin schemes, Edrei–Voiculescu theorem, inverse Vandermonde matrix,

*q*-deformation, skew Schur polynomials.

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