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 Mosc. Math. J., 2014, Volume 14, Number 1, Pages 121–160 (Mi mmj517)

The boundary of the Gelfand–Tsetlin graph: new proof of Borodin–Olshanski's formula, and its $q$-analogue

Leonid Petrovab

a Dobrushin Mathematics Laboratory, Kharkevich Institute for Information Transmission Problems, Moscow, Russia
b Department of Mathematics, Northeastern University, 360 Huntington ave., Boston, MA 02115, USA

Abstract: In their 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 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.

Key words and phrases: Gelfand–Tsetlin graph, trapezoidal Gelfand–Tsetlin schemes, Edrei–Voiculescu theorem, inverse Vandermonde matrix, $q$-deformation, skew Schur polynomials.

DOI: https://doi.org/10.17323/1609-4514-2014-14-1-121-160

Full text: http://www.mathjournals.org/.../2014-014-001-006.html
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Bibliographic databases:

MSC: 05E10, 22E66, 31C35, 46L65
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Citation: Leonid Petrov, “The boundary of the Gelfand–Tsetlin graph: new proof of Borodin–Olshanski's formula, and its $q$-analogue”, Mosc. Math. J., 14:1 (2014), 121–160

Citation in format AMSBIB
\Bibitem{Pet14} \by Leonid~Petrov \paper The boundary of the Gelfand--Tsetlin graph: new proof of Borodin--Olshanski's formula, and its $q$-analogue \jour Mosc. Math.~J. \yr 2014 \vol 14 \issue 1 \pages 121--160 \mathnet{http://mi.mathnet.ru/mmj517} \crossref{https://doi.org/10.17323/1609-4514-2014-14-1-121-160} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3221949} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000342789200006} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. I. Olshanskii, “Approximation of Markov Dynamics on the Dual Object of the Infinite-Dimensional Unitary Group”, Funct. Anal. Appl., 49:4 (2015), 289–300
2. V. Gorin, G. Panova, “Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory”, Ann. Probab., 43:6 (2015), 3052–3132
3. A. Bufetov, L. Petrov, “Law of large numbers for infinite random matrices over a finite field”, Selecta Math. (N.S.), 21:4 (2015), 1271–1338
4. A. Bufetov, V. Gorin, “Representations of classical Lie groups and quantized free convolution”, Geom. Funct. Anal., 25:3 (2015), 763–814
5. M. Ciucu, I. Fischer, “Lozenge tilings of hexagons with arbitrary dents”, Adv. in Appl. Math., 73 (2016), 1–22
6. V. Gorin, G. Olshanski, “A quantization of the harmonic analysis on the infinite-dimensional unitary group”, J. Funct. Anal., 270:1 (2016), 375–418
7. G. I. Olshanskii, “Extended Gelfand–Tsetlin Graph, Its $q$-Boundary, and $q$-B-Splines”, Funct. Anal. Appl., 50:2 (2016), 107–130
8. G. Olshanski, “The representation ring of the unitary groups and Markov processes of algebraic origin”, Adv. Math., 300 (2016), 544–615
9. G. Olshanski, “Markov dynamics on the dual object to the infinite-dimensional unitary group”, Probability and statistical physics in St. Petersburg, Proc. Sympos. Pure Math., 91, Amer. Math. Soc., Providence, RI, 2016, 373–394
10. S. Mkrtchyan, L. Petrov, “GUE corners limit of $q$-distributed lozenge tilings”, Electron. J. Probab., 22 (2017), 101
11. C. Cuenca, “Pieri integral formula and asymptotics of Jack unitary characters”, Sel. Math.-New Ser., 24:3 (2018), 2737–2789