The Gelfand-Tsetlin graph and Markov processes

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3 Citations (Scopus)

Abstract

The goal of the paper is to describe new connections between representation theory and algebraic combinatorics on one side, and probability theory on the other side. The central result is a construction, by essentially algebraic tools, of a family of Markov processes. The common state space of these processes is an infinite dimensional (but locally compact) space . It arises in representation theory as the space of indecomposable characters of the infinite-dimensional unitary group U(∞). Alternatively, can be defined in combinatorial terms as the boundary of the Gelfand-Tsetlin graph - an infinite graded graph that encodes the classical branching rule for characters of the compact unitary groups U(N). We also discuss two other topics concerning the Gelfand-Tsetlin graph: (1) Computation of the number of trapezoidal Gelfand-Tsetlin schemes (one could also say, the number of integral points in a truncated Gelfand-Tsetlin polytope). The formula we obtain is well suited for asymptotic analysis. (2) A degeneration procedure relating the Gelfand-Tsetlin graph to the Young graph by means of a new combinatorial object, the Young bouquet. At the end we discuss a few related works and further developments.

Original languageEnglish
Title of host publicationInvited Lectures
EditorsSun Young Jang, Young Rock Kim, Dae-Woong Lee, Ikkwon Yie, Young Rock Kim, Dae-Woong Lee, Ikkwon Yie
PublisherKYUNG MOON SA Co. Ltd.
Pages431-453
Number of pages23
ISBN (Electronic)9788961058070
Publication statusPublished - 2014
Externally publishedYes
Event2014 International Congress of Mathematicans, ICM 2014 - Seoul, Korea, Republic of
Duration: 13 Aug 201421 Aug 2014

Publication series

NameProceeding of the International Congress of Mathematicans, ICM 2014
Volume4

Conference

Conference2014 International Congress of Mathematicans, ICM 2014
Country/TerritoryKorea, Republic of
CitySeoul
Period13/08/1421/08/14

Keywords

  • Asymptotic representation theory
  • Feller Markov processes
  • Gelfand-Tsetlin graph
  • Infinitesimal generators
  • Representation ring
  • Symmetric functions
  • Young graph

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