Positive formula for Jack polynomials, Jack characters and proof of Lassalle's conjecture (2305.07966v1)

Abstract: We give an explicit formula for the power-sum expansion of Jack polynomials. We deduce it from a more general formula, which we provide here, that interprets Jack characters in terms of bipartite maps. We prove Lassalle's conjecture from 2008 on integrality and positivity of Jack characters in Stanley's coordinates and give a new formula for Jack polynomials using creation operators.

• The paper provides a positive formula for the power-sum expansion of Jack polynomials, resolving a conjecture by Lassalle.
• The study introduces a new differential calculus approach and connects layered map generating series to characterize Jack characters.
• Proving Lassalle's 2008 conjecture on Jack character properties provides new insights for algebraic combinatorics and computational methods.

An Overview of "Positive Formula for Jack Polynomials, Jack Characters and Proof of Lassalle's Conjecture"

This paper presents a comprehensive paper of Jack polynomials and Jack characters, culminating in the proof of Lassalle's conjecture regarding these mathematical constructs. The authors provide an explicit formula for the power-sum expansion of Jack polynomials, leveraging the broader context of bipartite maps and their combinatorial structures.

Key Contributions and Results

1. Power-Sum Expansion of Jack Polynomials: The paper delivers a precise combinatorial expression for Jack polynomials in the power-sum basis. This formulation directly addresses a longstanding question posited by Hanlon and offers a resolution to its more refined variant conjectured by Lassalle. The innovative approach interprets Jack characters in the framework of bipartite maps.
2. Proof of Lassalle's Conjecture: Lassalle's conjecture, articulated in 2008, suggested integrality and positivity properties of Jack characters when expressed in Stanley's coordinates. The authors not only prove this conjecture but also provide new insights into the mathematical underpinnings that interpolate between the enumerative properties of orientable and non-orientable graph maps.
3. Differential Calculus Approach and New Methods: A significant methodological contribution of the paper is its development of a differential calculus approach, which is instrumental in deriving the main results. This technique, complemented by algebraic characterizations and explorations of combinatorial aspects in the context of integrable systems, sheds new light on the structure and interpretations of Jack polynomials.
4. Generating Series and Layered Maps: The paper introduces generating series of layered maps, establishing a connection with Jack polynomials. By integrating this perspective, the paper efficiently characterizes Jack characters and elucidates their algebraic and combinatorial complexities.

Theoretical and Practical Implications

The theoretical contributions of this paper have profound implications for the domains of algebraic combinatorics, representation theory, and probability theory. By proving Lassalle's conjecture, the authors reinforce the understanding of the structure of Jack polynomials, which are pivotal in numerous mathematical physics models, including $\beta$-ensembles and the Calogero–Sutherland model.

Practically, the insights garnered about Jack characters can inform computational methods and algorithms designed for tasks involving symmetric functions. These applications are particularly relevant in fields requiring high-level polynomial evaluations and expansions.

Speculation on Future Developments in AI and Mathematics

The approaches and results expounded in this paper could inspire advanced computational techniques employing AI. For instance, optimization algorithms could integrate these mathematical insights to enhance symbolic computation frameworks, facilitating more robust and efficient handling of large polynomial structures.

Furthermore, as AI continues to intersect with advanced mathematics, the methodologies explored here might serve as a basis for developing machine learning models that can discern and predict complex combinatorial structures or patterns within datasets.

In summary, this paper not only resolves a significant mathematical conjecture but also expands the toolkit available to researchers dealing with symmetric functions and combinatorial mathematics. The implications of these findings are set to influence both theoretical research and practical computational applications in the field.