From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866911757001293824 |
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| author | Gorsky, Eugene Simental, José Vazirani, Monica |
| author_facet | Gorsky, Eugene Simental, José Vazirani, Monica |
| contents | In this note we explicitly construct an action of the rational Cherednik algebra $H_{1,m/n}(S_n,\mathbb{C}^n)$ corresponding to the permutation representation of $S_n$ on the $\mathbb{C}^{*}$-equivariant homology of parabolic Hilbert schemes of points on the plane curve singularity $\{x^{m} = y^{n}\}$ for coprime $m$ and $n$. We use this to construct actions of quantized Gieseker algebras on parabolic Hilbert schemes on the same plane curve singularity, and actions of the Cherednik algebra at $t = 0$ on the equivariant homology of parabolic Hilbert schemes on the non-reduced curve $\{y^{n} = 0\}.$ Our main tool is the study of the combinatorial representation theory of the rational Cherednik algebra via the subalgebra generated by Dunkl-Opdam elements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_14873 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra Gorsky, Eugene Simental, José Vazirani, Monica Representation Theory Algebraic Geometry Combinatorics In this note we explicitly construct an action of the rational Cherednik algebra $H_{1,m/n}(S_n,\mathbb{C}^n)$ corresponding to the permutation representation of $S_n$ on the $\mathbb{C}^{*}$-equivariant homology of parabolic Hilbert schemes of points on the plane curve singularity $\{x^{m} = y^{n}\}$ for coprime $m$ and $n$. We use this to construct actions of quantized Gieseker algebras on parabolic Hilbert schemes on the same plane curve singularity, and actions of the Cherednik algebra at $t = 0$ on the equivariant homology of parabolic Hilbert schemes on the non-reduced curve $\{y^{n} = 0\}.$ Our main tool is the study of the combinatorial representation theory of the rational Cherednik algebra via the subalgebra generated by Dunkl-Opdam elements. |
| title | From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra |
| topic | Representation Theory Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2004.14873 |