A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomials
In our previous paper, we gave a presentation of the torus-equivariant quantum K-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendiec...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| author | Toshiaki Maeno Satoshi Naito Daisuke Sagaki |
| author_facet | Toshiaki Maeno Satoshi Naito Daisuke Sagaki |
| author_sort | Toshiaki Maeno |
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| description | In our previous paper, we gave a presentation of the torus-equivariant quantum K-theory ring
$QK_{H}(Fl_{n+1})$
of the (full) flag manifold
$Fl_{n+1}$
of type
$A_{n}$
as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum K-theory ring
$QK_{H}(Fl_{n+1})$
under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant K-group of the semi-infinite flag manifold associated to
$SL_{n+1}(\mathbb {C})$
. |
| format | Article |
| id | doaj-art-8009e591e0d0472d866f2e24b203d4cc |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-8009e591e0d0472d866f2e24b203d4cc2025-01-30T05:12:53ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.147A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomialsToshiaki Maeno0Satoshi Naito1Daisuke Sagaki2https://orcid.org/0000-0002-9320-8147Department of Mathematics, Faculty of Science and Technology, Meijo University, 1-501 Shiogamaguchi, Tempaku-ku, Nagoya, 468-8502, Japan; E-mail:Department of Mathematics, Institute of Science Tokyo, 2-12-1 Oh-okayama, Meguro-ku, Tokyo, 152-8551, Japan; E-mail:Department of Mathematics, Institute of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571, JapanIn our previous paper, we gave a presentation of the torus-equivariant quantum K-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum K-theory ring $QK_{H}(Fl_{n+1})$ under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant K-group of the semi-infinite flag manifold associated to $SL_{n+1}(\mathbb {C})$ .https://www.cambridge.org/core/product/identifier/S2050509424001476/type/journal_article14N1514N3514M1505E1405E05 |
| spellingShingle | Toshiaki Maeno Satoshi Naito Daisuke Sagaki A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomials Forum of Mathematics, Sigma 14N15 14N35 14M15 05E14 05E05 |
| title | A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomials |
| title_full | A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomials |
| title_fullStr | A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomials |
| title_full_unstemmed | A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomials |
| title_short | A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomials |
| title_sort | presentation of the torus equivariant quantum k theory ring of flag manifolds of type a part ii quantum double grothendieck polynomials |
| topic | 14N15 14N35 14M15 05E14 05E05 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001476/type/journal_article |
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