Abstract

The goal of this thesis is to introduce and prove the $n!$ conjecture, this work is mainly based on the work of Mark Haiman from 1992 to 2001.\\
The $n!$ conjecture was for the first time approached to try to prove another conjecture, the positivity conjecture about the Kostka coefficients $K_{\lambda\mu}(q,t)$ which states that they belongs to the polynomial ring $\mathbb{N}[q,t]$.\\
It was known that the modules involved in the $n!$ conjecture are quotients of the ring $R_n$ of coinvariants for the action of $S_n$ on $\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n]$, denoted as $\mathbb{C}[\bold x, \bold y]$, and also that $R_n$ was isomorphic to the space of diagonal harmonics. Unfortunately, despite the computations suggesting that the dimension of $R_n$ should be $(n+1)^{n-1}$, proving it resulted very hard.\\
In the spring of 1992 Procesi and Haiman discussed the topic: Procesi suggested that the Hilbert scheme $H_n$ and what we now call the isospectral Hilbert scheme $X_n$ should be relevant to the determination of the dimension and character of $R_n$. Specifically, he observed that there is a natural map from $R_n$ to the ring of global functions on the scheme-theoretic fiber in $X_n$ over the origin in the symmetric power $S_n\mathbb{C}^2$, and with some luck this map could be an isomorphism! But let us make a step back and introducing the $n!$ conjecture properly.\\
Let $\mu=(\mu_{1},\mu_{2},\dots,\mu_{m})$ be a tuple of natural numbers such that $\sum_{i\in [m]}\mu_{i}=n$.\\
We define the Young Diagram associated to $\mu$ as the subset of $\mathbb{N}\times\mathbb{N}$ such that
\begin{equation*}
d(\mu)=\{(p,q)\,|\,p<\mu_{q+1}\}.
\end{equation*}
The conjecture states that if we take the alternating polynomial $\Delta_{\mu}$ defined as $\Delta_{\mu}=\det[x_i^{p_j}y_i^{q_j}]$ for $(p_j,q_j)\in \mu$ and we compute the space of all derivatives
\begin{equation*}
D_{\mu}=\mathbb{C}[\partial \bold x, \partial \bold y]\Delta_{\mu},
\end{equation*}
the dimension of $D_{\mu}$ is always $n!$.\\
Now it is important to see that there are three main topics to treat:
\begin{enumerate}
\item\label{intro_n!_conj} The $n!$ conjecture regarding the space $D_{\mu}$
\item \label{intro_pos_conj} The positivity conjecture regarding the Kostka coefficients $K_{\lambda\mu}(q,t)$
\item\label{intro_X_n} The isospectral Hilbert scheme of points $X_{n}$ and its natural map
\[
\rho:X_n\to H_n
\]
to the Hilbert scheme.
\end{enumerate}
In this introduction my goal is to make clear the connections between the points \ref{intro_n!_conj} and \ref{intro_X_n} as they are the main focus of this thesis. In Chapter $1$ the curious reader will also find a brief explaination of the connection between points \ref{intro_pos_conj} and \ref{intro_n!_conj}.\\
Let us begin with some mathematics.\\
The first thing that we can notice is that, if we take the ideal generated by $x^py^q$ for $(p,q)\notin \mu$ it is a monomial ideal, we will denote it by $I_{\mu}$.\\
There is a very nice property of monomial ideals: they are the fixed points of the action of $T=(\mathbb{C}^*)^{2}$ on the Hilbert scheme! Let's see why.\\
It's clear that $T$ acts on $\mathbb{C}^{2}$ sending $(a,b)$ to $(t_{1}a,t_{2}b)$, very similarly $T$ acts on $H_{n}=\text{Hilb}^{n}(\mathbb{C}^{2})$ by
\begin{equation*}
(t_{1},t_{2})I=(t_{1},t_{2})(f_{1}(x,y),\dots, f_{m}(x,y))\to (f_{1}(t_{1}x,t_{2}y),\dots,f_{m}(t_{1}x,t_{2}y)),
\end{equation*}
so if $I$ is monomial we can just factor the $t_{i}$ out without modifying anything.\\
The other important class of points of $H_{n}$ are the generic points denoted by $I=I(S)$, the ideals which vanishes on a specified finite set of distinct points $S\subseteq \mathbb{C}^2$ of cardinality $n$. In this very beautiful case $I$ is radical and $\mathbb{C}[x,y]/I$ is reduced and isomorphic to $\mathbb{C}^n$.
Intuitively we can think to $I$ as a set of $n$ points with multiplicity one and to $I_{\mu}$ as the origin with multiplicity $n$.\\
Notice that in $H_{n}$ the order of the points does not matter whether in $\mathbb{C}^{n}$ it does, so it is natural to consider the map
\begin{equation*}
\sigma:H_{n}\to \mathbb{C}^{n}/S_{n}
\end{equation*}
sending $I$ to the unordered n-tuple $(P_{1},\dots,P_{n})=V(I)$ of points. Notice that each $P\in V(I)$ appears in the $n$-tuple a number of time equal to its multiplicity.\\
Now $\sigma$ is called the \textit{Hilbert Chow Morphism} and it is a morphism of algebraic varieties and note that for $S=(P_{1},\dots,P_{n})$ all distinct in $\mathbb{C}^{n}/S_{n}$ there is only one ideal $I=I(S)\in H_{n}$ such that $\sigma(I)=S$, thus, giving the fact that the generic locus is dense in $H_{n}$ the map is \textit{birational}.\\
Later we will see that $H_{n}$ can also be described as a certain blowup of $\mathbb{C}^{n}/S_{n}$, so we can look at the Hilbert scheme of points as a resolution of the singularities of $\mathbb{C}^{n}/S_{n}$.\\
To recap let us look at the following diagram:
\begin{center}
\begin{tikzcd}
&\mathbb{C}^{2n}\arrow{d}\\
H_{n}\arrow{r}{\theta}& S_{n}\mathbb{C}^2
\end{tikzcd}
\end{center}
and notice that if we take a point $I(S)\in H_{n}$, we move it in $S_{n}\mathbb{C}^2$ and then we take the fiber in $\mathbb{C}^{2n}$ these fibers have lenght $n!$, in fact they are the sets of all possible orders of $n$ distinct points.\\
Unfortunately this argument does not hold for the monomial ideals $I_{\mu}$ thus we have to find another way to prove the conjecture.\\
An important property of finite flat morphism of schemes is that each fiber has the same lenght.\\
Now suppose that we can find a scheme lying above $H_{n}$ such that the map
\begin{equation*}
\rho:Y\to H_{n}
\end{equation*}
is flat and the fibers of a generic ideal $I$ have lenght $n!$, then we can use that property and conclude the proof!
Sadly the trivial choice of completing the above diagram with the fiber product does not work, the map is not flat.\\
Haiman's is to complete the diagram above with the reduced fiber product of $H_{n}$ and $\mathbb{C}^{2n}$ over $S_{n}\mathbb{C}^2$, we will call this space the \textit{isospectral Hilbert scheme} and denote it with $X_{n}$.
\begin{center}
\begin{tikzcd}
X_{n}\arrow{d}{\rho}\arrow{r}&\mathbb{C}^{2n}\arrow{d}\\
H_{n}\arrow{r}{\theta}& S_{n}\mathbb{C}^2
\end{tikzcd}
\end{center}
Now because $H_n$ is nonsingular and the projection $\rho: X_n \to H_n$ is finite, $X_n$ being Cohen-Macaulay is equivalent to $\rho$ being flat.\\
In particular the procedure is the following: we define the sheaf $B$ over the Hilbert scheme of points $H_n$ as the push-forward of $\mathcal{O}_{F}$ where $F$ is the universal family of $H_n$. Then we prove that we can see $X_n$ as $\spec(B^{\otimes n}/\mathcal{J})$ for a certain sheaf of ideals $\mathcal{J}$ and we prove that the ring
\[
B^{\otimes n}/\mathcal{J}\otimes_{\mathcal{O}_{H_n}}I_\mu
\]
is Cohen-Macaulay and Gorenstein.\\
There exists a very strong result (see \cite{emsalem1978geometrie}) proving that, up to isomorphism, a local Artinian $\mathbb{C}$-algebra is Gorenstein if and only if it is of the form $\mathbb{C}[\bold x]/J$ where
\[
J=\mathbb{C}[\partial\bold x]p,
\]
in other words $J$ is the vector space generated bya polynomial $p$ its partial derivarives of all orders.\\
So, proving that our ring $B^{\otimes n}/\mathcal{J}\otimes_{\mathcal{O}_{H_n}}I_\mu$ is Gorenstein it is actually equivalent to proving that it is of the form $\mathbb{C}[\bold x]/J$. Subsequently, with some computations, we manage to identify this ideal $J$, and with it, the dimension and the structure of our ring.\\
The process of proving $B^{\otimes n}/\mathcal{J}\otimes_{\mathcal{O}_{H_n}}I_\mu$ Gorenstein is very insidious, approximately it goes like that:
\begin{itemize}
\item We prove that $X_n$ is normal with a very ingenious argument using an algebraic structure called \textit{Polygraphs}.
\item We prove that the Gorenstein property is equivalent to the $n!$ conjecture, thus even the opposite implication works.
\item We prove the $n!$ conjecture by hand for $X_3$, then we start with an induction argument.
\item We use the equivalence: Cohen-Macaulay if and only if $\rho$ flat for normal varieties to suppose
\[
\rho:X_{n-1}\to H_{n-1}
\]
flat.
\item We use this ipothesis to prove that if $X_{n-1}$ is Gorenstein then $X_n$ is Gorenstein too.
\item $X_3$ is Gorenstein because the $n!$ conjecture holds, thus $X_n$ is Gorenstein and the $n!$ conjecture holds.
\end{itemize}
This thesis is organized into three chapters: the first one introduces the conjectures formally, gives an example of the $n!$ conjecture for small $n$ and delves into some element of representation theory of finite groups.
In the second chapter we dive into the algebraic geometry of the Hilbert scheme, the isospectral Hilbert scheme and we give a proof of the conjecture. During this proof we claim that the ideal
\[
J=\mathbb{C}[\bold x, \bold y]A
\]
where $A$ is the space of alternating polynomials is a free $\mathbb{C}[\bold y]$-module, the proof of this fact will take the entire third chapter.
Finally in the third chapter we introduce \textit{polygraphs}, a particular union of linear subspaces in $E^n\times E^l$ where $E=\mathbb{A}^2(\mathbb{C}).$\\
The motivation behind the name is that their constituent subspaces are the graphs of linear maps from $E^n$ to $E^l$.\\
The purpose of this section is to actually prove that the ring
\[
\mathcal{O}(Z(n,l))
\]
of the polygraph $Z(n,l)$ is a free $k[\bold y]$-module. Finally we find a map between this ring and $J$ taht concludes the argument.\\