Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ and $$q=0$$

Abstract

We examine the non-symmetric Macdonald polynomials E λ at q = 1, as well as the more general permuted-basement Macdonald polynomials. When q = 1, we show that E λ (x; 1, t) is symmetric and independent of t whenever λ is a partition. Furthermore, we show that, in general λ, 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 x, t, and the relative order of the entries in λ. We also examine the case q = 0, which gives rise to the so-called permutedbasement 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 basementpermuting operators as well as combinatorics on fillings, by applying results in a previous article by Per Alexandersson.Mathematics Subject Classification. 05E10, 05E05.

A Shuffle Theorem for Paths Under Any Line

“…Our (and , below) coincide with specializations of nonsymmetric Macdonald polynomials considered by Ion in [22, Theorem 4.8]. The twisted variants below are specializations of the ‘permuted basement’ nonsymmetric Macdonald polynomials studied (for ) by Alexandersson [1] and Alexandersson and Sawhney [2]. We also note that and have coefficients in and specialize at to Demazure characters and Demazure atoms, respectively.…”

“…Example 4.1.1. The picture below shows a tuple of skew diagrams 𝜈 = (𝜈 (1) , 𝜈 (2) ), with dashed lines indicating boxes of equal content and boxes numbered in reading order, along with a semistandard tableau T on 𝜈. 𝜈 = 𝜈 (1) 𝜈 (2) 1 2 4 7…”

A Shuffle Theorem for Paths Under Any Line

“…Our E λ (and F λ , below) coincide with specializations of non-symmetric Macdonald polynomials considered by Ion in [20,Theorem 4.8]. The twisted variants E σ λ below are specializations of the 'permuted basement' non-symmetric Macdonald polynomials studied (for GL l ) by Alexandersson [1] and Alexandersson and Sawhney [2].…”

“…In particular, b l ≤ b l−1 + 1, hence b l − v ≤ b l−1 + a l−1 + 1, and if equality holds, then b l−1 = ⌊p⌋, so σ(1) < σ (2). Using Lemma 4.3.4, and recalling that the definition (75) of F σ λ (x; q) is E σw 0 −λ (x; q), we have…”

Modified Macdonald polynomials and the multispecies zero-range process: I

“…Our approach involves new relations between the families of ASEP polynomials and of non-symmetric Macdonald polynomials at q = 1, building on the work of Alexandersson and Sawhney [AS19]. Among other results, we show that certain ratios of non-symmetric Macdonald polynomials become symmetric in the particular case q = 1.…”