Kavli Affiliate: Hiraku Nakajima
| First 5 Authors: Hiraku Nakajima, , , ,
| Summary:
We prove the conjecture by Gyenge, N’emethi and SzendrH{o}i in
arXiv:1512.06844, arXiv:1512.06848 giving a formula of the generating function
of Euler numbers of Hilbert schemes of points $operatorname{Hilb}^n(mathbb
C^2/Gamma)$ on a simple singularity $mathbb C^2/Gamma$, where $Gamma$ 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
$Gamma$ at $zeta = exp(frac{2pi i}{2(h^vee+1)})$ 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.
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