Schur Times Schubert via the Fomin-Kirillov Algebra
We study multiplication of any Schubert polynomial S[subscript w] by a Schur polynomial sλ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coeffic...
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Electronic Journal of Combinatorics
2014
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| Online Access: | http://hdl.handle.net/1721.1/89802 https://orcid.org/0000-0002-3964-8870 |
| _version_ | 1811083299806773248 |
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| author | Meszaros, Karola Panova, Greta Postnikov, Alexander |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Meszaros, Karola Panova, Greta Postnikov, Alexander |
| author_sort | Meszaros, Karola |
| collection | MIT |
| description | We study multiplication of any Schubert polynomial S[subscript w] by a Schur polynomial sλ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions λ, including hooks and the 2×2 box. We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of λ is a hook plus a box at the (2,2) corner. We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov.
This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients. |
| first_indexed | 2024-09-23T12:30:52Z |
| format | Article |
| id | mit-1721.1/89802 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:30:52Z |
| publishDate | 2014 |
| publisher | Electronic Journal of Combinatorics |
| record_format | dspace |
| spelling | mit-1721.1/898022022-10-01T09:26:41Z Schur Times Schubert via the Fomin-Kirillov Algebra Meszaros, Karola Panova, Greta Postnikov, Alexander Massachusetts Institute of Technology. Department of Mathematics Postnikov, Alexander We study multiplication of any Schubert polynomial S[subscript w] by a Schur polynomial sλ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions λ, including hooks and the 2×2 box. We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of λ is a hook plus a box at the (2,2) corner. We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov. This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients. National Science Foundation (U.S.) (Grant DMS-6923772) 2014-09-18T16:09:15Z 2014-09-18T16:09:15Z 2014-02 2013-08 Article http://purl.org/eprint/type/JournalArticle 1077-8926 http://hdl.handle.net/1721.1/89802 Meszaros, Karola, Greta Panova, and Alexander Postnikov. "Schur Times Schubert via the Fomin-Kirillov Algebra." Electronic Journal of Combinatorics, Volume 21, Issue 1 (2014). https://orcid.org/0000-0002-3964-8870 en_US http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p39 Electronic Journal of Combinatorics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Electronic Journal of Combinatorics Electronic Journal of Combinatorics |
| spellingShingle | Meszaros, Karola Panova, Greta Postnikov, Alexander Schur Times Schubert via the Fomin-Kirillov Algebra |
| title | Schur Times Schubert via the Fomin-Kirillov Algebra |
| title_full | Schur Times Schubert via the Fomin-Kirillov Algebra |
| title_fullStr | Schur Times Schubert via the Fomin-Kirillov Algebra |
| title_full_unstemmed | Schur Times Schubert via the Fomin-Kirillov Algebra |
| title_short | Schur Times Schubert via the Fomin-Kirillov Algebra |
| title_sort | schur times schubert via the fomin kirillov algebra |
| url | http://hdl.handle.net/1721.1/89802 https://orcid.org/0000-0002-3964-8870 |
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