Toda-Type Presentations for the Quantum K Theory of Partial Flag Varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912721561190400 |
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| author | Amini, Kamyar Huq-Kuruvilla, Irit Mihalcea, Leonardo C. Orr, Daniel Xu, Weihong |
| author_facet | Amini, Kamyar Huq-Kuruvilla, Irit Mihalcea, Leonardo C. Orr, Daniel Xu, Weihong |
| contents | We prove a determinantal, Toda-type, presentation for the equivariant K theory of a partial flag variety ${\rm Fl}(r_1, \dots, r_k;n)$. The proof relies on pushing forward the Toda presentation obtained by Maeno, Naito and Sagaki for the complete flag variety ${\rm Fl}(n)$, via Kato's ${\rm K}_T({\rm pt})$-algebra homomorphism from the quantum K ring of ${\rm Fl}(n)$ to that of ${\rm Fl}(r_1, \dots, r_k;n)$. Starting instead from the Whitney presentation for ${\rm Fl}(n)$, we show that the same pushforward technique gives a recursive formula for polynomial representatives of quantum K Schubert classes in any partial flag variety which do not depend on quantum parameters. In an appendix, we include another proof of the Toda presentation for the equivariant quantum K ring of ${\rm Fl}(n)$, following Anderson, Chen, and Tseng, which is based on the fact that the ${\rm K}$-theoretic $J$-function is an eigenfunction of the finite difference Toda Hamiltonians. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_07412 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Toda-Type Presentations for the Quantum K Theory of Partial Flag Varieties Amini, Kamyar Huq-Kuruvilla, Irit Mihalcea, Leonardo C. Orr, Daniel Xu, Weihong Algebraic Geometry Combinatorics Representation Theory 14M15, 14N35, 37K10, 05E05 We prove a determinantal, Toda-type, presentation for the equivariant K theory of a partial flag variety ${\rm Fl}(r_1, \dots, r_k;n)$. The proof relies on pushing forward the Toda presentation obtained by Maeno, Naito and Sagaki for the complete flag variety ${\rm Fl}(n)$, via Kato's ${\rm K}_T({\rm pt})$-algebra homomorphism from the quantum K ring of ${\rm Fl}(n)$ to that of ${\rm Fl}(r_1, \dots, r_k;n)$. Starting instead from the Whitney presentation for ${\rm Fl}(n)$, we show that the same pushforward technique gives a recursive formula for polynomial representatives of quantum K Schubert classes in any partial flag variety which do not depend on quantum parameters. In an appendix, we include another proof of the Toda presentation for the equivariant quantum K ring of ${\rm Fl}(n)$, following Anderson, Chen, and Tseng, which is based on the fact that the ${\rm K}$-theoretic $J$-function is an eigenfunction of the finite difference Toda Hamiltonians. |
| title | Toda-Type Presentations for the Quantum K Theory of Partial Flag Varieties |
| topic | Algebraic Geometry Combinatorics Representation Theory 14M15, 14N35, 37K10, 05E05 |
| url | https://arxiv.org/abs/2504.07412 |