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Mesures centrales pour les graphes multiplicatifs, représentations d’algèbres de Lie et polytopes des poids

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. It was shown in [11] and [12] that the conditioning of natural random Littelmann paths to stay in their corresponding Weyl chamber is controlled by central measures on this type of graphs. Using the K-theory of associated C∗-algebras, Handelman [8] 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 the Littelmann path model describing the iterated tensor product of an irreducible representation.
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Cédric Lecouvey, Pierre Tarrago. Mesures centrales pour les graphes multiplicatifs, représentations d’algèbres de Lie et polytopes des poids. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2020, 70 (6), pp.2361-2407. ⟨10.5802/aif.3350⟩. ⟨hal-03289869⟩

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