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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