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

By Alexei Borodin, Grigori Olshanski. 2011

63 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ff4a7ee6a8a46d76d797db946e5e2eed2db106a0

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In Applied Mathematics

1. Alexey I. Bufetov
2. Vadim Gorin
3. Grigori Olshanski
4. Alexei Borodin

In Discrete Mathematics and Combinatorics

1. Alexei Borodin
2. Grigori Olshanski
3. Alexey I. Bufetov
4. Vadim Gorin
5. L. Petrov
6. Ivan Corwin
7. Philippe Biane

In Geometry and Topology

1. Vadim Gorin
2. Alexey I. Bufetov
3. Grigori Olshanski
4. Alexei Borodin
5. L. Petrov
6. Ivan Corwin

Citations in the corpus

• Representations of classical Lie groups and quantized free convolution (2013, methodology): “sequence of normalized conventional characters of irreducible representations of S(N) and U(N), respectively, as N!1. This approach was further developed and generalized in [Bo2], [KOO], [OO], [OO2], [BO], [Pe2], [GP], leading, in particular, to classication theorems for characters of innite dimensi”
• Representations of the Infinite Symmetric Group (2016, methodology): “Our exposition is based on the works of Olshanski [87] and Kerov, Olshanski, and Vershik [67].”