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| Natura: | Preprint |
| Pubblicazione: |
2021
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| Accesso online: | https://arxiv.org/abs/2110.09985 |
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| _version_ | 1866910906739326976 |
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| author | Chow, Chi Hong |
| author_facet | Chow, Chi Hong |
| contents | We give a new proof of an unpublished result of Dale Peterson, proved by Lam and Shimozono, which identifies explicitly the structure constants, with respect to the quantum Schubert basis, for the $T$-equivariant quantum cohomology $QH^{\bullet}_T(G/P)$ of any flag variety $G/P$ with the structure constants, with respect to the affine Schubert basis, for the $T$-equivariant Pontryagin homology $H^T_{\bullet}(\mathcal{G}r)$ of the affine Grassmannian $\mathcal{G}r$ of $G$, where $G$ is any simple simply-connected complex algebraic group.
Our approach is to construct an $H_T^{\bullet}(pt)$-algebra homomorphism by Gromov-Witten theory and show that it is equal to Peterson's map. More precisely, the map is defined via Savelyev's generalized Seidel representations which can be interpreted as certain Gromov-Witten invariants with input $H^T_{\bullet}(\mathcal{G}r)\otimes QH_T^{\bullet}(G/P)$. We determine these invariants completely, in a way similar to how Fulton and Woodward did in their proof of quantum Chevalley formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_09985 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Peterson-Lam-Shimozono's theorem is an affine analogue of quantum Chevalley formula Chow, Chi Hong Algebraic Geometry Symplectic Geometry 14N35, 57T15 We give a new proof of an unpublished result of Dale Peterson, proved by Lam and Shimozono, which identifies explicitly the structure constants, with respect to the quantum Schubert basis, for the $T$-equivariant quantum cohomology $QH^{\bullet}_T(G/P)$ of any flag variety $G/P$ with the structure constants, with respect to the affine Schubert basis, for the $T$-equivariant Pontryagin homology $H^T_{\bullet}(\mathcal{G}r)$ of the affine Grassmannian $\mathcal{G}r$ of $G$, where $G$ is any simple simply-connected complex algebraic group. Our approach is to construct an $H_T^{\bullet}(pt)$-algebra homomorphism by Gromov-Witten theory and show that it is equal to Peterson's map. More precisely, the map is defined via Savelyev's generalized Seidel representations which can be interpreted as certain Gromov-Witten invariants with input $H^T_{\bullet}(\mathcal{G}r)\otimes QH_T^{\bullet}(G/P)$. We determine these invariants completely, in a way similar to how Fulton and Woodward did in their proof of quantum Chevalley formula. |
| title | Peterson-Lam-Shimozono's theorem is an affine analogue of quantum Chevalley formula |
| topic | Algebraic Geometry Symplectic Geometry 14N35, 57T15 |
| url | https://arxiv.org/abs/2110.09985 |