This article is cited in 11 scientific papers (total in 11 papers)
The boundary of the Gelfand–Tsetlin graph: new proof of Borodin–Olshanski's formula, and its $q$-analogue
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
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.
MSC: 05E10, 22E66, 31C35, 46L65
Received: September 17, 2012
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
\paper The boundary of the Gelfand--Tsetlin graph: new proof of Borodin--Olshanski's formula, and its $q$-analogue
\jour Mosc. Math.~J.
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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
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V. Gorin, G. Olshanski, “A quantization of the harmonic analysis on the infinite-dimensional unitary group”, J. Funct. Anal., 270:1 (2016), 375–418
G. I. Olshanskii, “Extended Gelfand–Tsetlin Graph, Its $q$-Boundary, and $q$-B-Splines”, Funct. Anal. Appl., 50:2 (2016), 107–130
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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
S. Mkrtchyan, L. Petrov, “GUE corners limit of $q$-distributed lozenge tilings”, Electron. J. Probab., 22 (2017), 101
C. Cuenca, “Pieri integral formula and asymptotics of Jack unitary characters”, Sel. Math.-New Ser., 24:3 (2018), 2737–2789
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