Seidel and Pieri products in cominuscule quantum K-theory
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arXiv
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| Format: | Preprint |
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2023
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| author | Buch, Anders S. Chaput, Pierre-Emmanuel Perrin, Nicolas |
| author_facet | Buch, Anders S. Chaput, Pierre-Emmanuel Perrin, Nicolas |
| contents | We prove a collection of formulas for products of Schubert classes in the quantum $K$-theory ring $QK(X)$ of a cominuscule flag variety $X$. This includes a $K$-theory version of the Seidel representation, stating that the quantum product of a Seidel class with an arbitrary Schubert class is equal to a single Schubert class times a power of the deformation parameter $q$. We also prove new Pieri formulas for the quantum $K$-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, and give a new proof of the known Pieri formula for the quantum $K$-theory of Grassmannians of type A. Our formulas have simple statements in terms of quantum shapes that represent the natural basis elements $q^d[{\mathcal O}_{X^u}]$ of $QK(X)$. Along the way we give a simple formula for $K$-theoretic Gromov-Witten invariants of Pieri type for Lagrangian Grassmannians, and prove a rationality result for the points in a Richardson variety in a symplectic Grassmannian that are perpendicular to a point in projective space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_05307 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Seidel and Pieri products in cominuscule quantum K-theory Buch, Anders S. Chaput, Pierre-Emmanuel Perrin, Nicolas Algebraic Geometry 14N35 (Primary) 19E08, 14N15, 14M15, 14E08 (Secondary) We prove a collection of formulas for products of Schubert classes in the quantum $K$-theory ring $QK(X)$ of a cominuscule flag variety $X$. This includes a $K$-theory version of the Seidel representation, stating that the quantum product of a Seidel class with an arbitrary Schubert class is equal to a single Schubert class times a power of the deformation parameter $q$. We also prove new Pieri formulas for the quantum $K$-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, and give a new proof of the known Pieri formula for the quantum $K$-theory of Grassmannians of type A. Our formulas have simple statements in terms of quantum shapes that represent the natural basis elements $q^d[{\mathcal O}_{X^u}]$ of $QK(X)$. Along the way we give a simple formula for $K$-theoretic Gromov-Witten invariants of Pieri type for Lagrangian Grassmannians, and prove a rationality result for the points in a Richardson variety in a symplectic Grassmannian that are perpendicular to a point in projective space. |
| title | Seidel and Pieri products in cominuscule quantum K-theory |
| topic | Algebraic Geometry 14N35 (Primary) 19E08, 14N15, 14M15, 14E08 (Secondary) |
| url | https://arxiv.org/abs/2308.05307 |