[Paper Review] Central measures on multiplicative graphs, representations of Lie algebras and weight polytopes

Cédric Lecouvey, Pierre Tarrago|arXiv (Cornell University)|Sep 1, 2016

Random Matrices and Applications11 references3 citations

TL;DR

This paper explicitly constructs a homeomorphism between central measures on multiplicative graphs associated with finite-dimensional representations of simple Lie algebras and the weight polytope of the representation, using Littelmann’s path model. It provides a parametrization of central measures via drifts of random Littelmann paths, leading to a complete characterization of harmonic and c-harmonic functions and a law of large numbers for conditioned random walks.

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.

Motivation & Objective

• To provide an explicit, representation-theoretic construction of the homeomorphism between central measures on multiplicative graphs and the weight polytope, independent of K-theory and C*-algebra methods.
• To characterize the minimal boundaries of central measures on infinite paths in the Weyl chamber and in the full space via drift parameters of Littelmann paths.
• To give a complete description of harmonic and c-harmonic functions for the Littelmann path model on multiplicative graphs.
• To establish a law of large numbers for random walks under central measures, linking path drifts to polytope geometry.
• To generalize Kerov-Vershik's framework beyond the defining representation of sl_n to arbitrary dominant weights of simple Lie algebras.

Proposed method

• Use of Littelmann’s path model to define random walks on the weight lattice via concatenation of paths in the representation’s weight set.
• Parametrization of central measures by assigning real drifts to simple roots, restricted to a subset [0,1]_δ^d defined by δ-admissible subsets of simple roots.
• Establishment of a bijection between extremal harmonic functions on the growth graph and algebra morphisms from the character algebra to R, nonnegative on the basis.
• Application of Weyl characters and generating functions to express the total weight of paths and derive the drift parametrization.
• Use of the generating function S_{x,nδ}(t) and normalization via s_δ(t) to define the measure and its drift dependence.
• Proof of the law of large numbers via convergence of path end positions to the drift vector, using the fact that s_δ(t) is strictly convex and minimized at t=1.

Experimental results

Research questions

• RQ1 How can the homeomorphism between central measures on multiplicative graphs and the weight polytope be made explicit without relying on C*-algebra K-theory?
• RQ2 What is the precise parametrization of central measures in terms of drifts of random Littelmann paths?
• RQ3 How are harmonic and c-harmonic functions on the path model characterized in terms of the drift parameters?
• RQ4 What is the limiting behavior of conditioned random walks under central measures, and how does it relate to the weight polytope?
• RQ5 Can the law of large numbers for such walks be derived from the path model and character theory?

Key findings

• The minimal boundary of central measures on infinite paths in the Weyl chamber Δ is explicitly parametrized by the set [0,1]_δ^d, which corresponds to δ-admissible subsets of simple roots.
• The homeomorphism between central measures on the full space and the weight polytope K(δ) is explicitly constructed using the drift vector t ∈ [0,1]_δ^d.
• The set of c-harmonic measures is identified as the level set of the function s_δ(t) = cZ, where Z = dim V(δ), and this set is non-empty only when c ≥ 1.
• The law of large numbers holds: almost surely, the end position of a random path under a central measure converges to the drift vector t scaled by the path length.
• The function s_δ(t) is strictly convex on R^d, with a unique minimum at t=1, and s_δ(1) = Z = dim V(δ), which implies that the minimal value of the drift function is achieved at the uniform drift.
• The support of any c-harmonic measure is contained in the level set {t ∈ [0,1]_δ^d | s_δ(t) = cZ}, and this set is precisely the minimal boundary of c-harmonic measures.