[Paper Review] Littelmann path model for geometric crystals, Whittaker functions on Lie groups and Brownian motion

Reda Chhaibi|arXiv (Cornell University)|Feb 4, 2013

Bayesian Methods and Mixture Models18 citations

TL;DR

This paper establishes a Littelmann path model for geometric crystals in complex semi-simple Lie groups, linking representation theory and probability by constructing a canonical measure via Wiener measure and total positivity. It derives Whittaker functions as Laplace transforms of this measure, showing they generalize characters in combinatorial crystals and recover the Duistermaat-Heckman measure and Littlewood-Richardson rules in the geometric setting.

ABSTRACT

Generally speaking, this thesis focuses on the interplay between the representations of Lie groups and probability theory. It subdivides into essentially three parts. In a first rather algebraic part, we construct a path model for geometric crystals in the sense of Berenstein and Kazhdan, for complex semi-simple Lie groups. We will mainly describe the algebraic structure, its natural morphisms and parameterizations. The theory of total positivity will play a particularly important role. Then, we anticipate on the probabilistic part by exhibiting a canonical measure on geometric crystals. It uses as ingredients the superpotential for the flag manifold and a measure invariant under the crystal actions. The image measure under the weight map plays the role of Duistermaat-Heckman measure. Its Laplace transform defines Whittaker functions, providing an interesting formula for all Lie groups. Then it appears clearly that Whittaker functions are to geometric crystals, what characters are to combinatorial crystals. The Littlewood-Richardson rule is also exposed. Finally we present the probabilistic approach that allows to find the canonical measure. It is based on the fundamental idea that the Wiener measure will induce the adequate measure on the algebraic structures through the path model. In the last chapter, we show how our geometric model degenerates to the continuous classical Littelmann path model and thus recover known results. For example, the canonical measure on a geometric crystal of highest weight degenerates into a uniform measure on a polytope, and recovers the parameterizations of continuous crystals.

Motivation & Objective

• To develop a Littelmann path model for geometric crystals in complex semi-simple Lie groups.
• To construct a canonical measure on geometric crystals using superpotentials and invariance under crystal actions.
• To show that the Laplace transform of this measure yields Whittaker functions across all Lie groups.
• To demonstrate that Whittaker functions play a role in geometric crystals analogous to characters in combinatorial crystals.
• To recover known results such as the Littlewood-Richardson rule and uniform measures on polytopes via degeneration to the classical continuous path model.

Proposed method

• Construct a path model for geometric crystals using the framework of Berenstein and Kazhdan, emphasizing total positivity.
• Define a canonical measure on geometric crystals using the superpotential on the flag manifold and invariance under crystal actions.
• Use the weight map to project the measure, identifying its image as a Duistermaat-Heckman-type measure.
• Derive Whittaker functions as the Laplace transform of this measure, establishing a universal formula for all Lie groups.
• Apply the Wiener measure on paths to induce the canonical measure on geometric crystals, linking probability and algebraic structure.
• Show degeneration of the geometric model to the continuous classical Littelmann path model, recovering known results.

Experimental results

Research questions

• RQ1 How can a Littelmann path model be extended to geometric crystals in complex semi-simple Lie groups?
• RQ2 What canonical measure arises naturally on geometric crystals using total positivity and invariance?
• RQ3 How do Whittaker functions emerge as Laplace transforms of this measure across all Lie groups?
• RQ4 In what way do Whittaker functions generalize characters in the context of geometric crystals?
• RQ5 How does the geometric model degenerate to the classical continuous path model, and what known results are recovered?

Key findings

• The canonical measure on geometric crystals is constructed using the superpotential on the flag manifold and invariance under crystal actions.
• The image of this measure under the weight map is identified as a Duistermaat-Heckman measure, generalizing known results in symplectic geometry.
• Whittaker functions are shown to be the Laplace transform of the canonical measure, providing a universal formula valid for all complex semi-simple Lie groups.
• The Whittaker function plays in geometric crystals the role analogous to characters in combinatorial crystals.
• The Littlewood-Richardson rule is recovered through the structure of the geometric crystal and its measure.
• Degeneration of the geometric model to the continuous classical Littelmann path model recovers uniform measures on polytopes and known parameterizations of continuous crystals.