Non-symmetric Macdonald polynomials and Demazure-Lusztig operators

Abstract

We extend the family non-symmetric Macdonald polynomials and define permuted-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 P λ are expressed as a sum of permuted-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 nonnegative 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.

Demazure crystals and the Schur positivity of Catalan functions

“…Theorem 2.3. For any tame labeled root ideal (Ψ, γ, w) with γ ∈ Z ℓ ≥0 , H(Ψ; γ; w) = π w x γ 1 1 Φπ s(n 1 )…”

“…A U q ( sl ℓ )-generalized Demazure crystal is a subset of a tensor product of highest weight crystals of the form 1 for some Λ 1 , . .…”

“…Our results apply to the type A nonsymmetric Macdonald polynomials at t = 0, E α (x; q, 0), a nonsymmetric generalization of the modified Hall-Littlewood polynomials. They were connected to affine Demazure characters by Sanderson [76] and the subject of recent results and conjectures on key positivity [1,2,4,5,6]. The t = 0 nonsymmetric Macdonald polynomials in other types have also received considerable attention [30,61,62,72].…”

“…The former quantity is not bounded below, so to make this induction valid we first handle the following "base case": suppose ℓ i=j γ i < 0 for some 2 ≤ j ≤ ℓ. Thus ℓ i=j−1 R(γ) i < 0, and so by Proposition 5.5 (iv), H(Ψ; γ; w a+1 ) = 0 = x γ 1 1 Φ H(R(Ψ); R(γ); w a ) . Next, the base case |Ψ| = 0 holds by Lemma 5.8 (it is here we need γ 1 ≥ 0):…”

“…Dating back to Sanderson [76], the E α (x; q, 0) are characters of certain U q ( sl ℓ )-Demazure crystals. This topic has recently regained popularity [1,2,4,5,6,61,62,72], and in particular Assaf-Gonzalez [5,6] gave a key positive formula for E α (x; q, 0). Our results yield a different key positive formula, which generalizes Lascoux's tableau formula for cocharge Kostka-Foulkes polynomials [53].…”