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Properties of Non-symmetric Macdonald Polynomials at q=1 and q=0
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
MIT,USA.
2019 (English)In: Annals of Combinatorics, ISSN 0218-0006, E-ISSN 0219-3094, Vol. 23, no 2, p. 219-239Article in journal (Refereed) Published
Abstract [en]

We examine the non-symmetric Macdonald polynomials E at q=1, as well as the more general permuted-basement Macdonald polynomials. When q=1, we show that E(x;1,t) is symmetric and independent of t whenever is a partition. Furthermore, we show that, in general , 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 . 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.

Place, publisher, year, edition, pages
Springer Publishing Company, 2019. Vol. 23, no 2, p. 219-239
Keywords [en]
Macdonald polynomials, Elementary symmetric functions, Key polynomials, Hall-Littlewood, Demazure characters, Factorization
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-254509DOI: 10.1007/s00026-019-00432-zISI: 000471710700001Scopus ID: 2-s2.0-85065717865OAI: oai:DiVA.org:kth-254509DiVA, id: diva2:1336621
Note

QC 20190709

Available from: 2019-07-09 Created: 2019-07-09 Last updated: 2019-07-29Bibliographically approved

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