Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866912498422120448 |
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| author | Fujita, Naoki |
| author_facet | Fujita, Naoki |
| contents | A toric degeneration of an irreducible variety is a flat degeneration to an irreducible toric variety. In the case of a flag variety, its toric degeneration with desirable properties induces degenerations of Richardson varieties to unions of irreducible closed toric subvarieties, called semi-toric degenerations. For instance, Morier-Genoud proved that Caldero's toric degenerations arising from string polytopes have this property. Semi-toric degenerations are closely related to Schubert calculus. Indeed, Kogan-Miller constructed semi-toric degenerations of Schubert varieties from Knutson-Miller's semi-toric degenerations of matrix Schubert varieties which give a geometric proof of the pipe dream formula of Schubert polynomials. In this paper, we focus on a toric degeneration of a flag variety arising from a cluster structure, and prove that it induces semi-toric degenerations of Richardson varieties. Our semi-toric degeneration can be regarded as a generalization of Morier-Genoud's and Kogan-Miller's semi-toric degenerations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_12731 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties Fujita, Naoki Algebraic Geometry Combinatorics Representation Theory Primary 14M15, Secondary 05E10, 13F60, 14M25 A toric degeneration of an irreducible variety is a flat degeneration to an irreducible toric variety. In the case of a flag variety, its toric degeneration with desirable properties induces degenerations of Richardson varieties to unions of irreducible closed toric subvarieties, called semi-toric degenerations. For instance, Morier-Genoud proved that Caldero's toric degenerations arising from string polytopes have this property. Semi-toric degenerations are closely related to Schubert calculus. Indeed, Kogan-Miller constructed semi-toric degenerations of Schubert varieties from Knutson-Miller's semi-toric degenerations of matrix Schubert varieties which give a geometric proof of the pipe dream formula of Schubert polynomials. In this paper, we focus on a toric degeneration of a flag variety arising from a cluster structure, and prove that it induces semi-toric degenerations of Richardson varieties. Our semi-toric degeneration can be regarded as a generalization of Morier-Genoud's and Kogan-Miller's semi-toric degenerations. |
| title | Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties |
| topic | Algebraic Geometry Combinatorics Representation Theory Primary 14M15, Secondary 05E10, 13F60, 14M25 |
| url | https://arxiv.org/abs/2110.12731 |