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

Alexei Borodin, Grigori Olshanski

Research output: Contribution to journalArticlepeer-review

32 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1738-1779
Number of pages42
JournalAdvances in Mathematics
Volume230
Issue number4-6
DOIs
Publication statusPublished - Jul 2012
Externally publishedYes

Keywords

  • Dual Schur functions
  • Gelfand-Tsetlin graph
  • Gelfand-Tsetlin schemes
  • Schur functions
  • Totally positive sequences
  • Unitary group characters

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