[Paper Review] Macdonald polynomials in superspace as eigenfunctions of commuting operators

Olivier Blondeau-Fournier, Patrick Desrosiers+2 more|arXiv (Cornell University)|Feb 17, 2012

Algebraic structures and combinatorial models24 references3 citations

TL;DR

This paper establishes the existence and uniqueness of Macdonald polynomials in superspace as common eigenfunctions of two families of commuting operators, resolving a long-standing conjecture. By constructing them via non-symmetric Macdonald polynomials and anticommuting monomials, the authors prove triangularity and orthogonality, uniquely characterizing the superpolynomials up to normalization.

ABSTRACT

A generalization of the Macdonald polynomials depending upon both commuting and anticommuting variables has been introduced recently. The construction relies on certain orthogonality and triangularity relations. Although many superpolynomials were constructed as solutions of highly over-determined system, the existence issue was left open. This is resolved here: we demonstrate that the underlying construction has a (unique) solution. The proof uses, as a starting point, the definition of the Macdonald superpolynomials in terms of the Macdonald non-symmetric polynomials via a non-standard (anti)symmetrization and a suitable dressing by anticommuting monomials. This relationship naturally suggests the form of two family of commuting operators that have the defined superpolynomials as their common eigenfunctions. These eigenfunctions are then shown to be triangular and orthogonal. Up to a normalization, these two conditions uniquely characterize these superpolynomials. Moreover, the Macdonald superpolynomials are found to be orthogonal with respect to a second (constant-term-type) scalar product and its norm is evaluated. The latter is shown to match (up to a q-power) the conjectured norm with respect to the original scalar product. Finally, we recall the super-version of the Macdonald positivity conjecture and present two new conjectures which both provide a remarkable relationship between the new (q,t)-Kostka coefficients and the usual ones.

Motivation & Objective

• To resolve the open problem of existence for Macdonald polynomials in superspace, which were previously defined only conjecturally.
• To establish a rigorous connection between Macdonald superpolynomials and non-symmetric Macdonald polynomials through a non-standard (anti)symmetrization and anticommuting monomial dressing.
• To characterize the superpolynomials as eigenfunctions of two commuting operator families, providing a constructive and unique definition.
• To prove orthogonality with respect to two scalar products, including a constant-term-type inner product, and evaluate the norm of the superpolynomials.
• To propose new conjectures relating generalized (q,t)-Kostka coefficients to the classical ones, extending the Macdonald positivity conjecture.

Proposed method

• Define Macdonald superpolynomials via a non-standard (anti)symmetrization of non-symmetric Macdonald polynomials, combined with a dressing by anticommuting monomials.
• Introduce two families of commuting operators (E₁ and E₂) whose common eigenfunctions are the superpolynomials, derived from the structure of the non-symmetric polynomials.
• Use the triangular decomposition of the superpolynomials in the monomial basis to prove uniqueness under the eigenfunction and triangularity conditions.
• Establish orthogonality with respect to the standard (q,t)-scalar product by proving self-adjointness of the operators and using duality.
• Introduce a second scalar product (constant-term type) and show that the superpolynomials are orthogonal under it, with a norm that matches the conjectured norm up to a q-power.
• Leverage the uniqueness of the solution to the eigenvalue problem under triangularity and orthogonality to confirm the existence of the superpolynomials.

Experimental results

Research questions

• RQ1 Do the Macdonald superpolynomials in superspace exist as solutions to the over-determined system defined by triangularity and orthogonality?
• RQ2 Can the Macdonald superpolynomials be characterized as eigenfunctions of a pair of commuting operators?
• RQ3 Is the norm of the Macdonald superpolynomials under the constant-term scalar product equal to the conjectured norm under the standard scalar product, up to a q-power?
• RQ4 What is the relationship between the generalized (q,t)-Kostka coefficients (defined via superpolynomials) and the classical (q,t)-Kostka coefficients?
• RQ5 Do the generalized (q,t)-Kostka coefficients exhibit positivity and satisfy a refined duality or symmetry property?

Key findings

• The Macdonald superpolynomials in superspace exist and are uniquely characterized by their triangular decomposition and orthogonality under the standard (q,t)-scalar product.
• The superpolynomials are shown to be eigenfunctions of two families of commuting operators E₁ and E₂, which are constructed from the non-symmetric Macdonald polynomials via a non-standard (anti)symmetrization.
• The norm of the Macdonald superpolynomials under the constant-term-type scalar product is computed and found to match the conjectured norm under the original scalar product up to a power of q.
• The superpolynomials are orthogonal with respect to both the standard (q,t)-scalar product and the constant-term scalar product, confirming their consistency across different inner products.
• Two new conjectures are proposed: one relating the generalized (q,t)-Kostaka coefficients to the classical ones via a duality, and another suggesting a positivity property for the generalized coefficients.
• The construction confirms the super-version of the Macdonald positivity conjecture and provides a framework for further exploration of generalized Kostka coefficients in supersymmetric settings.