Irreducible components of two-column $Δ$-Springer fibers
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910716393422848 |
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| author | Connor, Joshua P. Griffin, Sean T. Purohit, Kavish A. |
| author_facet | Connor, Joshua P. Griffin, Sean T. Purohit, Kavish A. |
| contents | The $Δ$-Springer fibers $Y_{n,λ,s}$, introduced by Levinson, Woo, and the second author, generalize Springer fibers for $\mathrm{GL}_n(\mathbb{C})$ and give a geometric interpretation of the of the Delta Conjecture from algebraic combinatorics (at $t=0$). We prove that all irreducible components of the $Δ$-Springer fiber $Y_{n,n-1}=Y_{n,(1^{n-1}),n-1}$ are smooth. In fact, we prove that any intersection of irreducible components of $Y_{n,n-1}$ is a smooth Hessenberg variety which has the structure of an iterated Grassmannian fiber bundle. We then give a presentation of the singular cohomology ring of each irreducible component of $Y_{n,n-1}$ and a combinatorial formula for the Poincaré polynomial of an arbitrary union of intersections of irreducible components in terms of arm and leg statistics on Dyck paths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17222 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Irreducible components of two-column $Δ$-Springer fibers Connor, Joshua P. Griffin, Sean T. Purohit, Kavish A. Combinatorics Algebraic Geometry The $Δ$-Springer fibers $Y_{n,λ,s}$, introduced by Levinson, Woo, and the second author, generalize Springer fibers for $\mathrm{GL}_n(\mathbb{C})$ and give a geometric interpretation of the of the Delta Conjecture from algebraic combinatorics (at $t=0$). We prove that all irreducible components of the $Δ$-Springer fiber $Y_{n,n-1}=Y_{n,(1^{n-1}),n-1}$ are smooth. In fact, we prove that any intersection of irreducible components of $Y_{n,n-1}$ is a smooth Hessenberg variety which has the structure of an iterated Grassmannian fiber bundle. We then give a presentation of the singular cohomology ring of each irreducible component of $Y_{n,n-1}$ and a combinatorial formula for the Poincaré polynomial of an arbitrary union of intersections of irreducible components in terms of arm and leg statistics on Dyck paths. |
| title | Irreducible components of two-column $Δ$-Springer fibers |
| topic | Combinatorics Algebraic Geometry |
| url | https://arxiv.org/abs/2411.17222 |