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June 2021 Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras
Hiraku Nakajima
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Kyoto J. Math. 61(2): 377-397 (June 2021). DOI: 10.1215/21562261-2021-0006

Abstract

We prove a recent conjecture by Gyenge, Némethi, and Szendrői giving a formula of the generating function of Euler numbers of Hilbert schemes of points HilbN(C2Γ) on a simple singularity C2Γ, where Γ is a finite subgroup of 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(2π12(h+1)) are always 1, which is a special case of an earlier conjecture by Kuniba. Here h is the dual Coxeter number. We also prove the claim, which was not known for E7, E8 before.

Dedication

To Fukaya san on the occasion of his 60th birthday

Citation

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Hiraku Nakajima. "Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras." Kyoto J. Math. 61 (2) 377 - 397, June 2021. https://doi.org/10.1215/21562261-2021-0006

Information

Received: 3 February 2020; Revised: 17 August 2020; Accepted: 1 September 2020; Published: June 2021
First available in Project Euclid: 29 March 2021

Digital Object Identifier: 10.1215/21562261-2021-0006

Subjects:
Primary: 14C05
Secondary: 14D21, 17B37

Rights: Copyright © 2021 by Kyoto University

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Vol.61 • No. 2 • June 2021
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