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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.00197 |
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| _version_ | 1866913631779684352 |
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| author | Gillespie, Maria Gorsky, Eugene Griffin, Sean T. |
| author_facet | Gillespie, Maria Gorsky, Eugene Griffin, Sean T. |
| contents | We introduce a variety $Y_{n,k}$, which we call the \textit{affine $Δ$-Springer fiber}, generalizing the affine Springer fiber studied by Hikita, whose Borel-Moore homology has an $S_n$ action and a bigrading that corresponds to the Delta Conjecture symmetric function $\mathrm{rev}_q\,ωΔ'_{e_{k-1}}e_n$ under the Frobenius character map. We similarly provide a geometric interpretation for the Rational Shuffle Theorem in the integer slope case $(km,k)$. The variety $Y_{n,k}$ has a map to the affine Grassmannian whose fibers are the $Δ$-Springer fibers introduced by Levinson, Woo, and the third author. Part of our proof of our geometric realization relies on our previous work on a Schur skewing operator formula relating the Rational Shuffle Theorem to the Delta Conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00197 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A geometric interpretation of the Delta Conjecture Gillespie, Maria Gorsky, Eugene Griffin, Sean T. Combinatorics Algebraic Geometry We introduce a variety $Y_{n,k}$, which we call the \textit{affine $Δ$-Springer fiber}, generalizing the affine Springer fiber studied by Hikita, whose Borel-Moore homology has an $S_n$ action and a bigrading that corresponds to the Delta Conjecture symmetric function $\mathrm{rev}_q\,ωΔ'_{e_{k-1}}e_n$ under the Frobenius character map. We similarly provide a geometric interpretation for the Rational Shuffle Theorem in the integer slope case $(km,k)$. The variety $Y_{n,k}$ has a map to the affine Grassmannian whose fibers are the $Δ$-Springer fibers introduced by Levinson, Woo, and the third author. Part of our proof of our geometric realization relies on our previous work on a Schur skewing operator formula relating the Rational Shuffle Theorem to the Delta Conjecture. |
| title | A geometric interpretation of the Delta Conjecture |
| topic | Combinatorics Algebraic Geometry |
| url | https://arxiv.org/abs/2501.00197 |