A stability phenomenon in Kazhdan-Lusztig combinatorics
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909571612672000 |
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| author | Creedon, Samuel Mazorchuk, Volodymyr |
| author_facet | Creedon, Samuel Mazorchuk, Volodymyr |
| contents | We prove that, when $n$ goes to infinity, the expression, with respect to the dual Kazhdan-Lusztig basis, of the product $\hat{\underline{H}}_x\underline{H}_y$ of elements of the dual and the usual Kazhdan-Lusztig bases in the Hecke algebra of the symmetric group $S_n$ stabilizes. As an application, we define the action of projective functors on the principal block of category $\mathcal{O}$ for $\mathfrak{sl}_\infty$ and show that the subcategory of finite length objects is stable under this action. As a bonus, we also prove that this latter block is Koszul, answering, for this block, a question from \cite{CP}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05931 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A stability phenomenon in Kazhdan-Lusztig combinatorics Creedon, Samuel Mazorchuk, Volodymyr Representation Theory We prove that, when $n$ goes to infinity, the expression, with respect to the dual Kazhdan-Lusztig basis, of the product $\hat{\underline{H}}_x\underline{H}_y$ of elements of the dual and the usual Kazhdan-Lusztig bases in the Hecke algebra of the symmetric group $S_n$ stabilizes. As an application, we define the action of projective functors on the principal block of category $\mathcal{O}$ for $\mathfrak{sl}_\infty$ and show that the subcategory of finite length objects is stable under this action. As a bonus, we also prove that this latter block is Koszul, answering, for this block, a question from \cite{CP}. |
| title | A stability phenomenon in Kazhdan-Lusztig combinatorics |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2504.05931 |