Central measures on multiplicative graphs, representations of Lie algebras and weight polytopes

Abstract : To each finite-dimensional representation of a simple Lie algebra is associated a multiplicative graph in the sense of Kerov and Vershik defined from the decomposition of its tensor powers into irreducible components. The conditioning of natural random Littelmann paths to stay in their corresponding Weyl chamber is then controlled by central measures on this type of graphs. Using the K-theory of associated C*-algebras, Handelman established a homeomorphism between the set of central measures on these multiplicative graphs and the weight polytope of the underlying representation. In the present paper, we make explicit this homeomorphism independently of Handelman's results by using Littelmann's path model. As a by-product we also get an explicit parametrization of the weight polytope in terms of drifts of random Littelmann paths. This explicit parametrization yields a complete description of harmonic and c-harmonic functions for this Littelmann paths model.
Complete list of metadatas

Cited literature [24 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01358385
Contributor : Cédric Lecouvey <>
Submitted on : Monday, May 27, 2019 - 6:05:19 PM
Last modification on : Thursday, June 6, 2019 - 1:35:48 AM

Files

HarmPiRev2305.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01358385, version 3
  • ARXIV : 1609.00138

Collections

Citation

Cedric Lecouvey, Pierre Tarrago. Central measures on multiplicative graphs, representations of Lie algebras and weight polytopes. 2016. ⟨hal-01358385v3⟩

Share

Metrics

Record views

24

Files downloads

20