[Paper Review] Non-symmetric Macdonald polynomials and Demazure-Lusztig operators

Per Alexandersson|arXiv (Cornell University)|Feb 16, 2016

Advanced Combinatorial Mathematics11 references21 citations

TL;DR

This paper introduces permuted-basement Macdonald polynomials as a generalization of non-symmetric Macdonald polynomials, showing they satisfy triangularity and behave well under Demazure–Lusztig operators. The key contribution is a unified formula expressing symmetric Macdonald polynomials as a positive sum over these polynomials, which generalizes known expansions into Hall–Littlewood polynomials and Demazure atoms, with explicit t-deformations of key polynomials and atoms emerging at q=0.

ABSTRACT

We extend the family non-symmetric Macdonald polynomials and define general-basement Macdonald polynomials. We show that these also satisfy a triangularity property with respect to the monomials bases and behave well under the Demazure-Lusztig operators. The symmetric Macdonald polynomials $J_λ$ are expressed as a sum of general-basement Macdonald polynomials via an explicit formula. By letting $q=0$, we obtain $t$-deformations of key polynomials and Demazure atoms and we show that the Hall--Littlewood polynomials expand positively into these. This generalizes a result by Haglund, Luoto, Mason and van Willigenburg. As a corollary, we prove that Schur polynomials decompose with non-negative coefficients into $t$-deformations of general Demazure atoms and thus generalizing the $t=0$ case which was previously known. This gives a unified formula for the classical expansion of Schur polynomials in Hall-Littlewood polynomials and the expansion of Schur polynomials into Demazure atoms.

Motivation & Objective

• To extend non-symmetric Macdonald polynomials by introducing a permutation parameter σ, forming a new family called permuted-basement Macdonald polynomials.
• To establish that these polynomials satisfy triangularity with respect to the monomial basis and behave well under Demazure–Lusztig operators.
• To provide an explicit expansion of symmetric Macdonald polynomials Pλ in terms of permuted-basement Macdonald polynomials.
• To generalize known results on Hall–Littlewood and Schur polynomial expansions by showing positivity in t-deformed atoms at q=0.
• To unify the classical expansion of Schur polynomials into Hall–Littlewood polynomials and into Demazure atoms via a single formula involving t-deformations.

Proposed method

• The construction uses non-attacking fillings of augmented diagrams with a fixed basement permutation σ and entries in {1,…,n}, defining combinatorial statistics such as inversions, coinversions, and major index.
• The polynomials are defined via a combinatorial formula involving q- and t-deformed weights based on leg and arm lengths of boxes in the diagram.
• The action of Demazure–Lusztig operators on the basement parameter σ is characterized, showing a simple transformation rule that enables recursive construction.
• The expansion of symmetric Macdonald polynomials is derived using the action of the operator ˜πσ on the non-symmetric version, leveraging known results from [HHL08].
• Specializations at q=0 yield t-deformations of Demazure atoms and Hall–Littlewood polynomials, with positivity established via combinatorial coefficients.
• The method relies on the theory of non-attacking fillings and affine Hecke algebra actions, particularly the use of ˜πσ and ˜θτ operators with braid and cancellation relations.

Experimental results

Research questions

• RQ1 How can non-symmetric Macdonald polynomials be generalized to include a permutation parameter σ on the basement, and what properties do these new polynomials satisfy?
• RQ2 Can the symmetric Macdonald polynomial Pλ be expressed as a positive linear combination of permuted-basement Macdonald polynomials?
• RQ3 What happens to the permuted-basement Macdonald polynomials under the specialization q=0, and how does this relate to Hall–Littlewood and Demazure polynomials?
• RQ4 Is there a unified formula that simultaneously generalizes the expansion of Schur polynomials into Hall–Littlewood polynomials and into Demazure atoms?
• RQ5 Do the Demazure–Lusztig operators act in a controlled way on the basement permutation σ, enabling recursive or combinatorial construction?

Key findings

• The symmetric Macdonald polynomial Pλ(x; q, t) expands positively as a sum over permuted-basement Macdonald polynomials with coefficients involving q- and t-deformed weights and the factor ttwinvπ(γ,σ).
• At q=0, the permuted-basement Macdonald polynomials become t-deformations of Demazure atoms, and Hall–Littlewood polynomials expand positively into these, generalizing a result by Haglund et al.
• Schur polynomials admit a positive expansion into t-deformed Demazure atoms, with coefficients given by Kostka–Foulkes polynomials Kλγ(t), unifying two classical expansions.
• The expansion formula (40) provides a single expression that reduces to both the Hall–Littlewood expansion and the Demazure atom expansion upon specialization.
• The action of Demazure–Lusztig operators on the basement parameter σ is shown to be simple and combinatorial, enabling recursive construction and proving the triangularity property.
• The paper proves that the number of cancellations in the operator expression ˜πσ˜θτxλ is exactly ttwinvπ(γ,σ), confirming the correct t-deformation factor in the final expression.