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 definedfrom the decomposition of its tensor powers into irreducible components. The conditioning of naturalrandom Littelmann paths to stay in their corresponding Weyl chamber is thencontrolled 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 theunderlying 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 theweight 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.
Keywords
Cite
@article{arxiv.1609.00138,
title = {Central measures on multiplicative graphs, representations of Lie algebras and weight polytopes},
author = {Cedric Lecouvey and Pierre Tarrago},
journal= {arXiv preprint arXiv:1609.00138},
year = {2017}
}
Comments
This version connects our results to those of Handelman and Pierce. It also gives a characterization of c-harmonic functions for the random Littelmann paths model considered