Framework
Recall that given a finite subgroup \(\Gamma\subset \mathrm{SL}_2(\mathbb{C})\) we have a McKay graph, which we denote by \(Q_\Gamma\). By choosing dimension vector \(v = n\delta\) and framing vector \(w = \rho_0\) (which adds a \(1\)-dim framing vertex to the node corresponding to the trivial representation of \(\Gamma\)),
when \(\theta=0\) this variety reduces to \[\mathfrak{M}_0(Q_\Gamma, v, w)=\mathrm{Sym}^n(\mathbb{C}^2/\Gamma),\] the \(n\)-th symmetric product of the orbifold \(\mathbb{C}^2/\Gamma\).
with stability \(\theta=(n, -1)\), \[\mathfrak{M}_\theta(Q_\Gamma, v, w)\cong \mathrm{Hilb}^n(\mathbb{C}^2/\Gamma)\] is isomorphic to the Hilbert scheme of points on the orbifold \(\mathbb{C}^2/\Gamma\).
For smooth \(X\), \(S^n X\) has \(p(n)\) stratums, where \(p(n)\) is the partition function. But here \(X=\mathbb{C}^2/\Gamma\) is no longer smooth, the number of stratums of \(\mathrm{Sym}^n(\mathbb{C}^2/\Gamma)\) is now equal to the stratums of \(\mathfrak{M}_0(Q_\Gamma, v, w)\), which has something to do with its subrepresentation structure, and for \(\theta=0\) all representations are semistable. (Question: how to determine the stratums in quiver varieties?)
The case \(\Gamma = \mathbb{Z}/2\)
The embedding of \(\mathbb{Z}_n\to \mathrm{SL}_2(\mathbb{C})\) is given by
\[ k\mapsto \begin{pmatrix} \omega_n^k & 0 \\ 0 & \omega_n^{-k} \end{pmatrix}, \]
in the case \(n=2\), we can compute the character table as
the McKay graph \(Q_\Gamma\) associated to \(\Gamma\) is
The points in \(\mathrm{Hilb}^n(\mathbb{C}^2/\mathbb{Z}_2)\)
The points in the hilbert scheme are in one to one correspondence with isomorphism classes of cyclic \(\mathbb{C}[x,y]^\Gamma = \mathbb{C}[x^2,xy,y^2]\)-modules together with a cyclic vector \((M,m)\) of dimension \(n\),
\[\begin{align*} \mathrm{Hilb}^n(\mathbb{C}^2/\Gamma) &\Leftrightarrow \{ \text{cyclic }(M,m) \mid \dim M = n \} / \cong \\ &\Leftrightarrow \{ I \subset \mathbb{C}[x,y]^\Gamma \mid \mathrm{codim\,}I = n \} \end{align*}\]
the correspondence from ideals to cyclic module pairs is given by
\[ I\mapsto (\mathbb{C}[x,y]^\Gamma/I, 1) \]
the correspondence from (iso-class of) modules to ideals is given by
\[ [(M,m)] \mapsto \mathrm{ann}(m). \]
The points in quiver variety \(\mathfrak{M}_0(Q_\Gamma, v, w)\)
Recall that the quiver variety is defined by
\[\mathfrak{M}_\theta(Q_\Gamma,v,w) := \mu^{-1}(0)/\!\!/_{\theta} \mathrm{GL}(v) \]
where \(\mu : \mathrm{Rep}(v,w)\to \bigoplus \mathrm{End}(v_i)\) is the moment map, and \(\Pi_0\) is the preprojective algebra. Here the quiver in question is
with dimension vector \(v=(1,n\delta) = (1, n, n)\).