A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866909115955019776 |
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| author | Lenart, Cristian Naito, Satoshi Sagaki, Daisuke |
| author_facet | Lenart, Cristian Naito, Satoshi Sagaki, Daisuke |
| contents | We give a Chevalley formula for an arbitrary weight for the torus-equivariant $K$-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum $K$-theory $QK_{T}(G/B)$ of an (ordinary) flag manifold $G/B$; this has been a longstanding conjecture about the multiplicative structure of $QK_{T}(G/B)$. In type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum $K$-theory $QK(SL_{n}/B)$; we also obtain very explicit information about the coefficients in the respective Chevalley formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_06143 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory Lenart, Cristian Naito, Satoshi Sagaki, Daisuke Combinatorics Algebraic Geometry Quantum Algebra Representation Theory Primary 14M15, 14N15, Secondary 14N10, 05E14, 17B37, 81R10 We give a Chevalley formula for an arbitrary weight for the torus-equivariant $K$-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum $K$-theory $QK_{T}(G/B)$ of an (ordinary) flag manifold $G/B$; this has been a longstanding conjecture about the multiplicative structure of $QK_{T}(G/B)$. In type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum $K$-theory $QK(SL_{n}/B)$; we also obtain very explicit information about the coefficients in the respective Chevalley formula. |
| title | A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory |
| topic | Combinatorics Algebraic Geometry Quantum Algebra Representation Theory Primary 14M15, 14N15, Secondary 14N10, 05E14, 17B37, 81R10 |
| url | https://arxiv.org/abs/2010.06143 |