Skip to content
DraconiticResearch
Search Sources API
189,280 studies31 sourcesUpdated 30 min ago
PreprintarXiv licencearXiv

Symmetric group actions on the coordinate ring of the lower unipotent group and the polynomial ring with quantum parameters

  • Tatsuya Horiguchi
Open at sourcePDFDOICite

Abstract

Givental-Kim and Ciocan-Fontanine gave an explicit presentation of the quantum cohomology ring of the flag variety. The author and Shirato introduced an algebraic generalization of their presentation in the context of the coordinate rings of regular nilpotent Hessenberg varieties. In particular, they connect the coordinate ring of the lower unipotent group in a general linear group and the polynomial ring with quantum parameters. In this paper we consider the dual of the connection and we see that the Plücker coordinates correspond to the quantizations of Schur polynomials. As an application of the connection, we construct an action of the symmetric group on the polynomial ring with quantum parameters. Using the symmetric group action, one can define the divided difference operators on the polynomial ring with quantum parameters. We study the quantizations of Schubert polynomials in relation to the divided difference operators.

Related

  1. Slow Entropy of Quasi-Unipotent Actions on S-Arithmetic Homogeneous Spaces
    Jasurbek Imomov
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  2. Directional measure rigidity for time-changed first row unipotent actions
    Davide Ravotti, Daren Wei
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  3. On the structure of polynomial multicorrelation sequences arising from nilpotent actions
    Jovan Andreevski, Felipe Hernández
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsDynamical systemsarXiv
  4. On skew-symmetric distributions and their use in Monte Carlo sampling algorithms: coordinate-free, Gibbs-style and manifold versions of the Barker proposal
    Minh Vu, Samuel Livingstone, Pantelis Samartsidis
    arXivPreprintCC BY 4.0Full textStatisticsarXiv
  5. Minimum-coordinate elimination on a simplex: polynomial first-step safety and coNP-complete correctness
    Arseny Kosachev
    arXivPreprintCC BY 4.0Full textOptimisation & controlarXiv

Cite

BibTeXRIS
BibTeX
@misc{horiguchi2026symmetric,
  title = {{Symmetric group actions on the coordinate ring of the lower unipotent group and the polynomial ring with quantum parameters}},
  author = {Tatsuya Horiguchi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28005},
  eprint = {2609.28005},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28005},
}
RIS
TY  - UNPB
TI  - Symmetric group actions on the coordinate ring of the lower unipotent group and the polynomial ring with quantum parameters
AU  - Tatsuya Horiguchi
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28005
UR  - https://arxiv.org/abs/2609.28005
AB  - Givental-Kim and Ciocan-Fontanine gave an explicit presentation of the quantum cohomology ring of the flag variety. The author and Shirato introduced an algebraic generalization of their presentation in the context of the coordinate rings of regular nilpotent Hessenberg varieties. In particular, they connect the coordinate ring of the lower unipotent group in a general linear group and the polynomial ring with quantum parameters. In this paper we consider the dual of the connection and we see that the Plücker coordinates correspond to the quantizations of Schur polynomials. As an application of the connection, we construct an action of the symmetric group on the polynomial ring with quantum parameters. Using the symmetric group action, one can define the divided difference operators on the polynomial ring with quantum parameters. We study the quantizations of Schubert polynomials in relation to the divided difference operators.
ER  -