Quantum bumpless pipe dreams
Schubert polynomials are polynomial representatives of Schubert classes in the cohomology of the complete flag variety and have a combinatorial formulation in terms of bumpless pipe dreams. Quantum double Schubert polynomials are polynomial representatives of Schubert classes in the torus-equivarian...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001129/type/journal_article |
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| _version_ | 1832086917237178368 |
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| author | Tuong Le Shuge Ouyang Leo Tao Joseph Restivo Angelina Zhang |
| author_facet | Tuong Le Shuge Ouyang Leo Tao Joseph Restivo Angelina Zhang |
| author_sort | Tuong Le |
| collection | DOAJ |
| description | Schubert polynomials are polynomial representatives of Schubert classes in the cohomology of the complete flag variety and have a combinatorial formulation in terms of bumpless pipe dreams. Quantum double Schubert polynomials are polynomial representatives of Schubert classes in the torus-equivariant quantum cohomology of the complete flag variety, but no analogous combinatorial formulation had been discovered. We introduce a generalization of the bumpless pipe dreams called quantum bumpless pipe dreams, giving a novel combinatorial formula for quantum double Schubert polynomials as a sum of binomial weights of quantum bumpless pipe dreams. We give a bijective proof for this formula by showing that the sum of binomial weights satisfies a defining transition equation. |
| format | Article |
| id | doaj-art-4bbc8276667344f797d4c8e4847f76ee |
| 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-4bbc8276667344f797d4c8e4847f76ee2025-02-06T09:14:53ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.112Quantum bumpless pipe dreamsTuong Le0https://orcid.org/0009-0002-0900-4983Shuge Ouyang1Leo Tao2Joseph Restivo3Angelina Zhang4Dept. of Mathematics, University of Michigan, Ann Arbor, MI, United StatesDept. of Mathematics, University of Michigan, Ann Arbor, MI, United States; E-mail:Dept. of Mathematics, University of Michigan, Ann Arbor, MI, United States; E-mail:Dept. of Mathematics, University of Michigan, Ann Arbor, MI, United States; E-mail:Dept. of Mathematics, University of Michigan, Ann Arbor, MI, United States; E-mail:Schubert polynomials are polynomial representatives of Schubert classes in the cohomology of the complete flag variety and have a combinatorial formulation in terms of bumpless pipe dreams. Quantum double Schubert polynomials are polynomial representatives of Schubert classes in the torus-equivariant quantum cohomology of the complete flag variety, but no analogous combinatorial formulation had been discovered. We introduce a generalization of the bumpless pipe dreams called quantum bumpless pipe dreams, giving a novel combinatorial formula for quantum double Schubert polynomials as a sum of binomial weights of quantum bumpless pipe dreams. We give a bijective proof for this formula by showing that the sum of binomial weights satisfies a defining transition equation.https://www.cambridge.org/core/product/identifier/S2050509424001129/type/journal_article05E05 |
| spellingShingle | Tuong Le Shuge Ouyang Leo Tao Joseph Restivo Angelina Zhang Quantum bumpless pipe dreams Forum of Mathematics, Sigma 05E05 |
| title | Quantum bumpless pipe dreams |
| title_full | Quantum bumpless pipe dreams |
| title_fullStr | Quantum bumpless pipe dreams |
| title_full_unstemmed | Quantum bumpless pipe dreams |
| title_short | Quantum bumpless pipe dreams |
| title_sort | quantum bumpless pipe dreams |
| topic | 05E05 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001129/type/journal_article |
| work_keys_str_mv | AT tuongle quantumbumplesspipedreams AT shugeouyang quantumbumplesspipedreams AT leotao quantumbumplesspipedreams AT josephrestivo quantumbumplesspipedreams AT angelinazhang quantumbumplesspipedreams |