A raising operator formula for Macdonald polynomials (2307.06517v1)

Abstract: We give an explicit raising operator formula for the modified Macdonald polynomials $\tilde{H}{\mu }(X;q,t)$, which follows from our recent formula for $\nabla$ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions $\tilde{H}^{1,n}(X;q,t)$ that we call $1,n$-Macdonald polynomials, which reduce to a scalar multiple of $\tilde{H}{\mu}(X;q,t)$ when $n=1$. We conjecture that the coefficients of $1,n$-Macdonald polynomials in terms of Schur functions belong to $\mathbb{N}[q,t]$, generalizing Macdonald positivity.

• The paper derives an explicit raising operator formula for modified Macdonald polynomials, addressing a longstanding gap in symmetric function theory.
• It employs rigorous algebraic techniques with the Haglund-Haiman-Loehr formula and the ∇ operator to express these polynomials as weighted sums of LLT polynomials.
• The results impact combinatorial interpretations and suggest novel conjectures on coefficient positivity in 1,n-Macdonald polynomials, paving the way for future research.

A Raising Operator Formula for Macdonald Polynomials

The paper focuses on explicating a raising operator formula for modified Macdonald polynomials, denoted as $H_\mu(X;q,t)$. The work stands on recent advances related to the application of the $\nabla$ operator on LLT (Lascoux, Leclerc, and Thibon) polynomials along with the Haglund-Haiman-Loehr formula that describes modified Macdonald polynomials as combinations of LLT polynomials.

Theoretical Contributions

The authors provide an explicit raising operator formula for these polynomials, a contribution that fills an existing gap in the literature. Previous expressions for other symmetric functions, such as Hall-Littlewood polynomials, are valuable due to their reliance on raising operators—a methodology that had been conspicuously absent for Macdonald polynomials until this work.

The formula derived here not only applies to traditional Macdonald polynomials but also extends to a family of symmetric functions called 1,n-Macdonald polynomials. These functions reduce to scalar multiples of $\mu(X;q,t)$ when $n=1$. They further conjecture that the coefficients of these 1,n-Macdonald polynomials, expressed in terms of Schur functions, belong to $\mathbb{N}[q,t]$, expanding the existing conjectured Macdonald positivity property.

Methodology

The paper employs rigorous algebraic manipulations and applies the Haglund-Haiman-Loehr formula, which facilitates expressing Macdonald polynomials as weighted sums of LLT polynomials. They show that the $\nabla$ operator, when applied to an LLT polynomial, aids in establishing eigenfunction properties necessary for their results. By scrutinizing these eigenfunction relations, the authors derive formulas that extend Macdonald polynomial theory.

Results and Implications

Strong numerical results lie in the establishment of a complete and easily applicable formula for the modified Macdonald polynomials, including specializations and generalizations for column diagrams. The raised hypothesis that the 1,n-Macdonald polynomials' coefficients are in $\mathbb{N}[q,t]$ suggests a potential new avenue in the combinatorial contexts of Macdonald polynomials, which might impact the broader field of algebraic combinatorics.

The implications are manifold:

1. Combinatorial Interpretations: This leads to deeper insights into how symmetric functions can be interpreted in terms of roots and weights.
2. Algebraic Structures: Understanding the raising operator in the context of Macdonald polynomials might shine a light on other functions within the same algebraic hierarchy.
3. Further Research: Given the conjecture about coefficients being in $\mathbb{N}[q,t]$, that area is ripe for exploration, potentially generalizing the applications of these polynomials further into geometric and representation theory.

Future Directions

This research paves the way for further analysis into polynomial expansions within other algebraic frameworks, possibly broadening into other operators analogous to the raising operator. Moreover, computational advancements, inspired by these theoretical models, might yield practical algorithms relevant in symbolic computation or even physical models.

The insights in this paper present a pathway to explore unsolved questions surrounding Macdonald positivity and open potential cross-disciplinary applications, notably if the conjectured properties can be verified across symmetric functions universally. This provides a firm ground for future theoretical developments and their implications in pure and applied mathematics.