Abstract
We examine the non-symmetric Macdonald polynomials mathrm {E}_lambda at q=1, as well as the more general permuted-basement Macdonald polynomials. When q=1, we show that mathrm {E}_lambda (mathbf {x};1,t) is symmetric and independent of t whenever lambda is a partition. Furthermore, we show that, in general lambda , this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of t, and the non-symmetric part only depends on mathbf {x}, t, and the relative order of the entries in lambda . We also examine the case q=0, which gives rise to the so-called permuted-basement t-atoms. We prove expansion properties of these t-atoms, and, as a corollary, prove that Demazure characters (key polynomials) expand positively into permuted-basement atoms. This complements the result that permuted-basement atoms are atom-positive. Finally, we show that the product of a permuted-basement atom and a Schur polynomial is again positive in the same permuted-basement atom basis. Haglund, Luoto, Mason, and van Willigenburg previously proved this property for the identity basement and the reverse identity basement, so our result can be seen as an interpolation (in the Bruhat order) between these two results. The common theme in this project is the application of basement-permuting operators as well as combinatorics on fillings, by applying results in a previous article by Per Alexandersson.
Highlights
The non-symmetric Macdonald polynomials Eλ(x; q, t) were introduced by Macdonald and Opdam in [19,22]
We only consider the type A for which there is a combinatorial rule, discovered by Haglund et al [11]. These non-symmetric Macdonald polynomials specialize to the Demazure characters, Kλ, at q = t = 0, they are affine Demazure characters, see [14]
Since a priori Eσλ(x; 1, t) is only a rational function in t, this seems like a difficult challenge
Summary
The non-symmetric Macdonald polynomials Eλ(x; q, t) were introduced by Macdonald and Opdam in [19,22]. We only consider the type A for which there is a combinatorial rule, discovered by Haglund et al [11] These non-symmetric Macdonald polynomials specialize to the Demazure characters, Kλ, (or key polynomials) at q = t = 0 (or at t = 0), they are affine Demazure characters, see [14]. In [4], various factorization properties of non-symmetric Macdonald polynomials are observed experimentally (in particular, the specialization q = u−2, t = u) in the last section of the article. Equation (1.2) is a generalization of the fact that key polynomials and permuted-basement atoms expand positively into Demazure atoms, see e.g. By taking τ = ω0, we see that key polynomials expand positively into permuted-basement Demazure atoms: Kγ(x) =. CσγαAσα(x), α : par(α)=par(γ) where cσγα ∈ {0, 1}
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