The fixed point locus of the smooth Jordan quiver variety under the action of the finite subgroups of $\mathrm{SL}_2(\mathbb{C})$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916957791453184 |
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| author | Paegelow, Raphaël |
| author_facet | Paegelow, Raphaël |
| contents | In this article, we study the decomposition into irreducible components of the fixed point locus under the action of $Γ$ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ of the smooth Nakajima quiver variety of the Jordan quiver. The quiver variety associated with the Jordan quiver is either isomorphic to the punctual Hilbert scheme in $\mathbb{C}^2$ or to the Calogero-Moser space. We describe the irreducible components using quiver varieties of McKay's quiver associated with the finite subgroup $Γ$ and we have given a general combinatorial model of the indexing set of these irreducible components in terms of certain elements of the root lattice of the affine Lie algebra associated with $Γ$. Finally, we prove that for every projective, symplectic resolution of a wreath product singularity, there exists an irreducible component of the fixed point locus of the punctual Hilbert scheme in the plane that is isomorphic to the resolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The fixed point locus of the smooth Jordan quiver variety under the action of the finite subgroups of $\mathrm{SL}_2(\mathbb{C})$ Paegelow, Raphaël Representation Theory Algebraic Geometry 16G20, 05E14, 14B05, 14C05, 17B67 In this article, we study the decomposition into irreducible components of the fixed point locus under the action of $Γ$ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ of the smooth Nakajima quiver variety of the Jordan quiver. The quiver variety associated with the Jordan quiver is either isomorphic to the punctual Hilbert scheme in $\mathbb{C}^2$ or to the Calogero-Moser space. We describe the irreducible components using quiver varieties of McKay's quiver associated with the finite subgroup $Γ$ and we have given a general combinatorial model of the indexing set of these irreducible components in terms of certain elements of the root lattice of the affine Lie algebra associated with $Γ$. Finally, we prove that for every projective, symplectic resolution of a wreath product singularity, there exists an irreducible component of the fixed point locus of the punctual Hilbert scheme in the plane that is isomorphic to the resolution. |
| title | The fixed point locus of the smooth Jordan quiver variety under the action of the finite subgroups of $\mathrm{SL}_2(\mathbb{C})$ |
| topic | Representation Theory Algebraic Geometry 16G20, 05E14, 14B05, 14C05, 17B67 |
| url | https://arxiv.org/abs/2407.01114 |