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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2412.20863 |
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| _version_ | 1866910766498578432 |
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| author | Graham, William Larson, Scott Joseph |
| author_facet | Graham, William Larson, Scott Joseph |
| contents | We study the torus-equivariant cohomology of weighted flag varieties, and prove a positivity property in the equivariant cohomology and Chow groups of weighted flag varieties, analogous to the non-weighted positivity proved in [Graham 2001]. Our result strengthens and generalizes the positivity proved for weighted Grassmannians by [Abe-Matsumura 2015]. The positivity property is expressed in terms of weighted roots, which are used to describe weights of torus equivariant curves in weighted flag varieties. This provides a geometric interpretation of the parameters used in [Abe-Matsumura 2015]. We approach weighted flag varieties from a uniform Lie-theoretic point of view, providing a more general definition than has appeared previously, and prove other general results about weighted flag varieties in this setting, including a Borel presentation of the equivariant cohomology. In addition, we generalize some results obtained for weighted Grassmannians or more generally type $A$ ([Abe-Matsumura 2015], [Azam-Nazir-Qureshi 2020]); in particular, we obtain a weighted Chevalley formula, descriptions of restrictions to fixed points, the GKM description of the cohomology, and a weighted Chevalley formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positivity in Weighted Flag Varieties Graham, William Larson, Scott Joseph Algebraic Geometry Representation Theory 55N91 (Primary), 14L24 (Secondary), 14C15 (Secondary) We study the torus-equivariant cohomology of weighted flag varieties, and prove a positivity property in the equivariant cohomology and Chow groups of weighted flag varieties, analogous to the non-weighted positivity proved in [Graham 2001]. Our result strengthens and generalizes the positivity proved for weighted Grassmannians by [Abe-Matsumura 2015]. The positivity property is expressed in terms of weighted roots, which are used to describe weights of torus equivariant curves in weighted flag varieties. This provides a geometric interpretation of the parameters used in [Abe-Matsumura 2015]. We approach weighted flag varieties from a uniform Lie-theoretic point of view, providing a more general definition than has appeared previously, and prove other general results about weighted flag varieties in this setting, including a Borel presentation of the equivariant cohomology. In addition, we generalize some results obtained for weighted Grassmannians or more generally type $A$ ([Abe-Matsumura 2015], [Azam-Nazir-Qureshi 2020]); in particular, we obtain a weighted Chevalley formula, descriptions of restrictions to fixed points, the GKM description of the cohomology, and a weighted Chevalley formula. |
| title | Positivity in Weighted Flag Varieties |
| topic | Algebraic Geometry Representation Theory 55N91 (Primary), 14L24 (Secondary), 14C15 (Secondary) |
| url | https://arxiv.org/abs/2412.20863 |