Type AIII orbits in the affine flag variety of type A
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912960621838336 |
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| author | Tong, Kam Hung |
| author_facet | Tong, Kam Hung |
| contents | Matsuki and Oshima introduced the notion of clans, which are incomplete matchings with positive or negative signs on isolated vertices. They discovered that clans parametrise $K$-orbits in the flag varieties for classical linear groups, where $K$ is a fixed point subgroup of an involution in the same classical linear group. In this work we investigate the affine version of these orbits. For a field $\Bbbk$ with characteristic not equal to two, we construct an explicit bijection between the $\textsf{GL}_p(\Bbbk(\hspace{-0.5mm}(t)\hspace{-0.5mm})) \times \textsf{GL}_q(\Bbbk(\hspace{-0.5mm}(t)\hspace{-0.5mm}))$-orbits in the affine flag variety and certain objects called affine $(p,q)$-clans. These affine $(p,q)$-clans can be concretely interpreted as involutions in the affine permutation group with positive or negative signs on fixed points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16269 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Type AIII orbits in the affine flag variety of type A Tong, Kam Hung Representation Theory 14L30, 20G05 Matsuki and Oshima introduced the notion of clans, which are incomplete matchings with positive or negative signs on isolated vertices. They discovered that clans parametrise $K$-orbits in the flag varieties for classical linear groups, where $K$ is a fixed point subgroup of an involution in the same classical linear group. In this work we investigate the affine version of these orbits. For a field $\Bbbk$ with characteristic not equal to two, we construct an explicit bijection between the $\textsf{GL}_p(\Bbbk(\hspace{-0.5mm}(t)\hspace{-0.5mm})) \times \textsf{GL}_q(\Bbbk(\hspace{-0.5mm}(t)\hspace{-0.5mm}))$-orbits in the affine flag variety and certain objects called affine $(p,q)$-clans. These affine $(p,q)$-clans can be concretely interpreted as involutions in the affine permutation group with positive or negative signs on fixed points. |
| title | Type AIII orbits in the affine flag variety of type A |
| topic | Representation Theory 14L30, 20G05 |
| url | https://arxiv.org/abs/2501.16269 |