The boundary of the Gelfand–Tsetlin graph: A new approach

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 inf...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2012-07, Vol.230 (4-6), p.1738-1779
Main Authors: Borodin, Alexei, Olshanski, Grigori
Format: Article
Language:English
Subjects:
Dual Schur functions
Gelfand–Tsetlin graph
Gelfand–Tsetlin schemes
Schur functions
Totally positive sequences
Unitary group characters
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Summary: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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2012.04.005