Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras
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arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866914549551071232 |
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| author | Nakajima, Hiraku |
| author_facet | Nakajima, Hiraku |
| contents | We prove the conjecture by Gyenge, Némethi and Szendrői in arXiv:1512.06844, arXiv:1512.06848 giving a formula of the generating function of Euler numbers of Hilbert schemes of points $\operatorname{Hilb}^n(\mathbb C^2/Γ)$ on a simple singularity $\mathbb C^2/Γ$, where $Γ$ is a finite subgroup of $\mathrm{SL}(2)$. We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with $Γ$ at $ζ= \exp(\frac{2πi}{2(h^\vee+1)})$ are always $1$, which is a special case of a conjecture by Kuniba [Kun93]. Here $h^\vee$ is the dual Coxeter number. We also prove the claim, which was not known for $E_7$, $E_8$ before. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_03834 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras Nakajima, Hiraku Algebraic Geometry High Energy Physics - Theory Quantum Algebra Representation Theory We prove the conjecture by Gyenge, Némethi and Szendrői in arXiv:1512.06844, arXiv:1512.06848 giving a formula of the generating function of Euler numbers of Hilbert schemes of points $\operatorname{Hilb}^n(\mathbb C^2/Γ)$ on a simple singularity $\mathbb C^2/Γ$, where $Γ$ is a finite subgroup of $\mathrm{SL}(2)$. We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with $Γ$ at $ζ= \exp(\frac{2πi}{2(h^\vee+1)})$ are always $1$, which is a special case of a conjecture by Kuniba [Kun93]. Here $h^\vee$ is the dual Coxeter number. We also prove the claim, which was not known for $E_7$, $E_8$ before. |
| title | Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras |
| topic | Algebraic Geometry High Energy Physics - Theory Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2001.03834 |