Borodin, Alexei and Olshanski, Grigori (2012) The boundary of the Gelfand–Tsetlin graph: A new approach. Advances in Mathematics, 230 (4-6). pp. 1738-1779. ISSN 0001-8708 http://resolver.caltech.edu/CaltechAUTHORS:20120713-143937716
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The Gelfand–Tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups. The boundary of the Gelfand–Tsetlin graph has at least three incarnations — as a discrete potential theory boundary, as the set of finite indecomposable characters of the infinite-dimensional unitary group, and as the set of doubly infinite totally positive sequences. An old deep result due to Albert Edrei and Dan Voiculescu provides an explicit description of the boundary; it can be realized as a region in an infinite-dimensional coordinate space. The paper contains a novel approach to the Edrei–Voiculescu theorem. It is based on a new explicit formula for the number of semi-standard Young tableaux of a given skew shape (or of Gelfand–Tsetlin schemes of trapezoidal shape). The formula is obtained via the theory of symmetric functions, and new Schur-like symmetric functions play a key role in the derivation.
|Additional Information:||© 2012 Elsevier Inc. Received 11 October 2011; accepted 8 April 2012; Available online 16 May 2012. Communicated by Andrei Zelevinsky. A.B. was partially supported by NSF-grant DMS-1056390. G.O. was partially supported by a grant from Simons Foundation (Simons—IUM Fellowship), the RFBR-CNRS grant 10-01- 93114, and the project SFB 701 of Bielefeld University.|
|Subject Keywords:||Gelfand–Tsetlin graph; Gelfand–Tsetlin schemes; Unitary group characters; Totally positive sequences; Schur functions; Dual Schur functions|
|Official Citation:||Alexei Borodin, Grigori Olshanski, The boundary of the Gelfand–Tsetlin graph: A new approach, Advances in Mathematics, Volume 230, Issues 4–6, July–August 2012, Pages 1738-1779, ISSN 0001-8708, 10.1016/j.aim.2012.04.005. (http://www.sciencedirect.com/science/article/pii/S0001870812001442)|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Aucoeur Ngo|
|Deposited On:||13 Jul 2012 22:00|
|Last Modified:||13 Jul 2012 22:00|
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