Lecture given by Paul Johnson, notes taken by Edward Pearce, 01/10/2019, 12pm-1pm at The University of Sheffield.
| Topology of \(\mathrm{Hilb}(S)\) | Representation of some algebra | Quiver varieties | Partitions |
|---|---|---|---|
| Cohomology \(H^{\ast}\) | Heisenberg algebra | ||
| Homology \(H_{\ast}\) | Virasoro algebra | ||
| K-theory \(K\) | Hall algebra | ||
| Derived Category \(\mathcal{D}\) |
For a partition \(\lambda=(5,3,3,2,1,1)\), the size of \(\lambda\), denoted \(|\lambda|\), is the sum of its parts, here 15, and the length of \(\lambda\), denoted \(\ell(\lambda)\), is the number of parts, here 6.
Let \(\mathcal{P}\) denote the set of all partitions, and \(\mathcal{P}(n)\) denote the set of partitions of (size) \(n\).
\[\sum_{\lambda\in\mathcal{P}}q^{|\lambda|}t^{\ell(\lambda)}=\prod_{m\ge1}\frac{1}{1-tq^{m}}=:\frac{1}{(t;q)_{\infty}}\]
where we call \((t;q)_{\infty}\) the Pochammer symbol.
We define the arm and leg of a cell/box in a partition, and the hook length of a cell as \(h(\square)=1+a(\square)+l(\square)\).
From the representation theory of symmetric groups, we get the hook length formula
\[\dim\lambda=\frac{|\lambda|!}{\prod_{\square\in\lambda}h(\square)}\]
Define the \(r\)-hook dimension of a partition \(\lambda\) to be
\[h_{r}(\lambda):=|\{\square\in\lambda:a(\square)>0, h(\square)\equiv0\bmod r\}|\]
We call a partition \(\lambda\) an $r$-core if \(r\) does not divide \(h(\square)\), i.e. \(r\nmid h(\square)\), for any \(\square\in\lambda\). Consequently, for \(\lambda\) an \(r\)-core, \(h_{r}(\lambda)=0\).
The generating function involving \(h_{r}(\lambda)\) can be expressed using the following product formula
\[\sum_{\lambda\in\mathcal{P}}q^{|\lambda|}t^{h_{r}(\lambda)}=\prod_{m\ge1}\frac{(1-q^{rm})^{r}}{1-q^{m}}\cdot\frac{1}{(1-q^{rm}t^{m-1})(1-q^{rm}t^{m})^{r-1}}\]
where the factor \(\frac{(1-q^{rm})^{r}}{1-q^{m}}\) corresponds to the generating function for \(r\)-cores.
Let \(0 < k < r\) be coprime integers, and define the \(k/r\)-hook dimension of a partition \(\lambda\) to be
\[h_{k/r}(\lambda):=|\{\square\in\lambda:a(\square)>0, ka(\square)-l(\square)-1\equiv0\bmod r)\}|\]
This is a generalization of the previous hook dimension since \(h_{(r-1)/r}=h_{r}\).
Define the \(k/r\)-hook dimension generating function to be
\[G_{k/r}=\sum_{\lambda\in\mathcal{P}}q^{|\lambda|}t^{h_{k/r}(\lambda)}\]
Gusein-Zade, Luengo, and Melle-Hernandez conjectured a product formula for the following case:
\[G_{1/3}=\frac{1}{(q;q^{3}t)_{\infty}}\cdot\frac{1}{(q^{2}t;q^{3}t)_{\infty}}\cdot\frac{1}{(q^{3};q^{3}t)_{\infty}}=\big(\frac{1}{1-q}\frac{1}{1-q^{2}t}\frac{1}{1-q^{3}}\big)\big(\frac{1}{1-q^{4}t}\frac{1}{1-q^{5}t^{2}}\frac{1}{1-q^{6}t}\big)\ldots\]
This summer Johnson and Rennemo proved a general product formula for \(G_{k/r}\).
Definition: \(\mathrm{Hilb}_{n}(\mathbb{C}^{2})=\{I\vartriangleleft R:\dim_{\mathbb{C}}R/I=n\}\) where \(R=\mathbb{C}[x, y]\).
If \(I\) is a reduced ideal, then we can interpret \(R/I\) as functions vanishing on \(n\) points in \(\mathbb{C}^{2}\). Therefore we may identify the reduced Hilbert scheme of \(n\) points on the complex plane with the smooth locus of \(\mathbb{C}^{2n}/S_{n}\). That is,
\[\mathrm{Hilb}_{n}^{\text{red}}(\mathbb{C}^{2})\cong\big(\mathbb{C}^{2n}\setminus\{p_{i}=p_{j}\}\big)/S_{n}=\big(\mathbb{C}^{2n}/S_{n}\big)_{\text{smooth}}\]
If points collide, then \(\mathbb{C}^{2n}/S_{n}\) becomes singular, and \(\mathrm{Hilb}\) fixes this by taking a resolution of singularities.
Consider the family of ideals
\[I_{t}=(x,y)\cap(x-at,y-bt)=(x^{2}-axt,xy-bxt,xy-ayt,y^{2}-byt)\]
which represent two points in \(\mathbb{C}^{2}\) which get closer and closer as $t$ tends to $0$.
If we naively take the limit \(t=0\), we would obtain \(I_{0}=(x^{2},xy,y^{2})\), but then \(\dim_{\mathbb{C}}R/I_{0}=3\ne2\), so \(I_{0}\notin\mathrm{Hilb}_{2}(\mathbb{C}^{2})\).
However, since \(ayt-bxt=(ay-bx)t\in I_{t}\) for all $t$ it follows that \(ay-bx\in I_{t}\) for all $t$, and indeed for \(I_{0}'=(x^{2},xy,y^{2}, ay-bx)\) we have \(\dim_{\mathbb{C}}R/I_{0}'=2\) and so \(I_{0}'\in\mathrm{Hilb}_{2}(\mathbb{C}^{2})\).
Therefore, in the case of points colliding, the Hilbert scheme retains extra information (the tangent direction of collision) which resolves the singularities along the diagonal of \(\mathbb{C}^{2n}/S_{n}\).
\(\pi:\mathrm{Hilb}_{n}(\mathbb{C}^{2})\to\mathbb{C}^{2n}/S_{n}\) is a crepent resolution of singularities, which means that \(\pi^{\ast}(K_{\mathbb{C}^{2n}/S_{n}})=K_{\mathrm{Hilb}_{n}(\mathbb{C}^{2})}\). In other words, the pullback along \(\pi\) of the canonical bundle of \(\mathbb{C}^{2n}/S_{n}\) is equal to the canonical bundle of \(\mathrm{Hilb}_{n}(\mathbb{C}^{2})\).
The Euler characteristic of \(\mathrm{Hilb}_{n}(\mathbb{C}^{2})\) is equal to the number of partitions of size \(n\). That is
\[\chi(\mathrm{Hilb}_{n}(\mathbb{C}^{2}))=p(n)\]
An algebraic torus \(T=(\mathbb{C}^{\times})^{2}\) acts on the complex plane \(\mathbb{C}^{2}\) and hence on \(\mathrm{Hilb}_{n}(\mathbb{C}^{2})\).
The fixed points of this action are the monomial ideals in \(\mathbb{C}[x,y]\), which are of the form \(I=(x^{a_{i}}y^{b_{i}})\) for a finite set of non-negative integer pairs \((a_{i},b_{i})\), and these in turn are in bijection with the set of partitions of size \(n\).
Using Białynicki-Birula decomposition (algebraic Morse theory), we can compute
\[\sum_{k}h_{k}(\mathrm{Hilb}_{n}(\mathbb{C}^{2}))\cdot t^{k}=\sum_{\text{fixed points partitions}}t^{f(\lambda)}\]
where \(f(\lambda)\) encodes data about the tangent space \(T_{\lambda}\mathrm{Hilb}_{n}(\mathbb{C}^{2})\) to \(\mathrm{Hilb}_{n}(\mathbb{C}^{2})\) at \(\lambda\) as a $T$-representation.
Ellingrud-Stromme proved that
\[\sum_{k,n}h_{k}(\mathrm{Hilb}_{n}(\mathbb{C}^{2}))t^{k}q^{n}=\prod_{m\ge1}\frac{1}{1-q^{m}t^{2m-2}}\]
Gottsche proved that for $S$ any smooth quasiprojective surface, we have
\[\sum_{k,n}h_{k}(\mathrm{Hilb}_{n}(S))t^{k}q^{n}=\prod_{\text{2 basis of }H_{0}(S)}\frac{1}{(qt^{\deg2};q^{m}t^{\alpha})_{\infty}}\]
whilst noting that \(S\) has only even cohomology.
The formula from the main theorem resembles Gottsche's formula with $r$ cohomology classes - some are in \(h_{0}\) and some are in \(h_{2}\).
\(G_{k/r}\) is a formula for
\[\sum_{n,k}h_{k}(\mathrm{Hilb}_{n}(\big[\mathbb{C}^{2}/\mathbb{Z}_{r}\big]))q^{n}t^{k}\]
where \(\mathbb{Z}_{r}\) acts on \(\mathbb{C}^{2}\) linearly via \(\left(\begin{array}{cc}e^{\frac{2\pi i}{r}} & 0\\0 & e^{\frac{2\pi i}{r}}\end{array}\right)\).