Author: Eiko

Tags: quiver variety, Hilbert scheme, ADE singularity

Framework

Recall that given a finite subgroup $\mathrm{\Gamma }\subset {\mathrm{SL}}_2(\mathbb{C})$ we have a McKay graph, which we denote by $Q_{\mathrm{\Gamma }}$. 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 $\mathrm{\Gamma }$),

• when $\theta =0$ this variety reduces to $${\mathfrak{M}}_0(Q_{\mathrm{\Gamma }},v,w)={\mathrm{Sym}}^n({\mathbb{C}}^2/\mathrm{\Gamma }),$$ the $n$-th symmetric product of the orbifold ${\mathbb{C}}^2/\mathrm{\Gamma }$.

• with stability $\theta =(n,-1)$, $${\mathfrak{M}}_{\theta }(Q_{\mathrm{\Gamma }},v,w)\cong {\mathrm{Hilb}}^n({\mathbb{C}}^2/\mathrm{\Gamma })$$ is isomorphic to the Hilbert scheme of points on the orbifold ${\mathbb{C}}^2/\mathrm{\Gamma }$.

For smooth $X$, $S^{n}X$ has $p(n)$ stratums, where $p(n)$ is the partition function. But here $X={\mathbb{C}}^2/\mathrm{\Gamma }$ is no longer smooth, the number of stratums of ${\mathrm{Sym}}^n({\mathbb{C}}^2/\mathrm{\Gamma })$ is now equal to the stratums of ${\mathfrak{M}}_0(Q_{\mathrm{\Gamma }},v,w)$, 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 $\mathrm{\Gamma }=\mathbb{Z}/2$

The embedding of ${\mathbb{Z}}_n\to {\mathrm{SL}}_2(\mathbb{C})$ is given by

$$k\mapsto \left(\begin{array}{cc}{\omega }_n^k& 0\\ 0& {\omega }_n^{-k}\end{array}\right),$$

in the case $n=2$, we can compute the character table as

the McKay graph $Q_{\mathrm{\Gamma }}$ associated to $\mathrm{\Gamma }$ 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 $\mathbb{C}[x,y{]}^{\mathrm{\Gamma }}=\mathbb{C}[x^2,xy,y^2]$-modules together with a cyclic vector $(M,m)$ of dimension $n$,

$$\begin{array}{rl}{\mathrm{Hilb}}^n({\mathbb{C}}^2/\mathrm{\Gamma })& \iff \{\text{cyclic }(M,m)\mid \dim M=n\}/\cong \\ & \iff \{I\subset \mathbb{C}[x,y{]}^{\mathrm{\Gamma }}\mid \mathrm{codim}I=n\}\end{array}$$

• the correspondence from ideals to cyclic module pairs is given by

  $$I\mapsto (\mathbb{C}[x,y{]}^{\mathrm{\Gamma }}/I,1)$$

• the correspondence from (iso-class of) modules to ideals is given by

  $$[(M,m)]\mapsto \mathrm{ann}(m).$$

The points in quiver variety ${\mathfrak{M}}_0(Q_{\mathrm{\Gamma }},v,w)$

Recall that the quiver variety is defined by

$${\mathfrak{M}}_{\theta }(Q_{\mathrm{\Gamma }},v,w):={\mu }^{-1}(0)/{/}_{\theta }\mathrm{GL}(v)$$

where $\mu :\mathrm{Rep}(v,w)\to ⨁\mathrm{End}(v_i)$ is the moment map, and ${\mathrm{\Pi }}_0$ is the preprojective algebra. Here the quiver in question is

with dimension vector $v=(1,n\delta )=(1,n,n)$.