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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


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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