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Article Dans Une Revue Annales de l'Institut Fourier Année : 2019

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

Résumé

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.
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Dates et versions

hal-01358385 , version 1 (31-08-2016)
hal-01358385 , version 2 (17-01-2017)
hal-01358385 , version 3 (27-05-2019)

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Citer

Cedric Lecouvey, Pierre Tarrago. Central measures on multiplicative graphs, representations of Lie algebras and weight polytopes. Annales de l'Institut Fourier, 2019. ⟨hal-01358385v3⟩
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