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

Cedric Lecouvey (1) , Pierre Tarrago (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

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 controlled by central measures on this type of graphs. In this paper we characterize all the central measures on these multiplicative graphs and explain how they can be easily parametrized by the weight polytope of the underlying representation. We also get an explicit parametrization of this weight polytope by the drifts of random Littelmann paths.

Domaines: Théorie des représentations [math.RT]; Combinatoire [math.CO]; Probabilités [math.PR]

Cedric Lecouvey, Pierre Tarrago. Central measures on multiplicative graphs, representations of Lie algebras and weight polytopes. 2016. ⟨hal-01358385v1⟩