Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ q = 1 and $$q=0$$ q = 0
Abstract We examine the non-symmetric Macdonald polynomials $$\mathrm {E}_\lambda $$ E...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer International Publishing
2021
|
| Online Access: | https://hdl.handle.net/1721.1/131597 |
| _version_ | 1826211885152731136 |
|---|---|
| author | Alexandersson, Per Sawhney, Mehtaab |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Alexandersson, Per Sawhney, Mehtaab |
| author_sort | Alexandersson, Per |
| collection | MIT |
| description | Abstract
We examine the non-symmetric Macdonald polynomials
$$\mathrm {E}_\lambda $$
E
λ
at
$$q=1$$
q
=
1
, as well as the more general permuted-basement Macdonald polynomials. When
$$q=1$$
q
=
1
, we show that
$$\mathrm {E}_\lambda (\mathbf {x};1,t)$$
E
λ
(
x
;
1
,
t
)
is symmetric and independent of t whenever
$$\lambda $$
λ
is a partition. Furthermore, we show that, in general
$$\lambda $$
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
$$\mathbf {x}$$
x
, t, and the relative order of the entries in
$$\lambda $$
λ
. We also examine the case
$$q=0$$
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. |
| first_indexed | 2024-09-23T15:12:38Z |
| format | Article |
| id | mit-1721.1/131597 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:12:38Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1315972023-03-15T17:27:33Z Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ q = 1 and $$q=0$$ q = 0 Alexandersson, Per Sawhney, Mehtaab Massachusetts Institute of Technology. Department of Mathematics Abstract We examine the non-symmetric Macdonald polynomials $$\mathrm {E}_\lambda $$ E λ at $$q=1$$ q = 1 , as well as the more general permuted-basement Macdonald polynomials. When $$q=1$$ q = 1 , we show that $$\mathrm {E}_\lambda (\mathbf {x};1,t)$$ E λ ( x ; 1 , t ) is symmetric and independent of t whenever $$\lambda $$ λ is a partition. Furthermore, we show that, in general $$\lambda $$ 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 $$\mathbf {x}$$ x , t, and the relative order of the entries in $$\lambda $$ λ . We also examine the case $$q=0$$ 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. 2021-09-20T17:28:54Z 2021-09-20T17:28:54Z 2019-05-11 2020-06-26T13:28:25Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131597 PUBLISHER_CC en https://doi.org/10.1007/s00026-019-00432-z Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Alexandersson, Per Sawhney, Mehtaab Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ q = 1 and $$q=0$$ q = 0 |
| title | Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ q = 1 and $$q=0$$ q = 0 |
| title_full | Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ q = 1 and $$q=0$$ q = 0 |
| title_fullStr | Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ q = 1 and $$q=0$$ q = 0 |
| title_full_unstemmed | Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ q = 1 and $$q=0$$ q = 0 |
| title_short | Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ q = 1 and $$q=0$$ q = 0 |
| title_sort | properties of non symmetric macdonald polynomials at q 1 q 1 and q 0 q 0 |
| url | https://hdl.handle.net/1721.1/131597 |
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