Schur Times Schubert via the Fomin-Kirillov Algebra
Name
Meszaros-2014-Schur times Schubert.pdf
Size
341.53 KB
Format
Adobe PDF
Checksum (MD5)
8948acd9152a5979f35b1fc6d9a89947
Author(s)
Meszaros, Karola
Panova, Greta
Postnikov, Alexander
Date Issued
February 2014
Journal
Electronic Journal of Combinatorics
Publisher
Electronic Journal of Combinatorics
Citation
Meszaros, Karola, Greta Panova, and Alexander Postnikov. "Schur Times Schubert via the Fomin-Kirillov Algebra." Electronic Journal of Combinatorics, Volume 21, Issue 1 (2014).
Version
Final published version
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
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.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.37236/3659
