Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study the isospectral Hilbert scheme $X_{n}$, defined as the reduced fiber product of $(\mathbb{C}^{2})^{n}$ with the Hilbert scheme $H_{n}$ of points in the plane $\mathbb{C}^{2}$, over the symmetric power $S^{n}\mathbb{C}^{2} = (\mathbb{C}^{2})^{n}/S_{n}$. By a theorem of Fogarty, $H_{n}$ is smooth. We prove that $X_{n}$ is normal, Cohen-Macaulay and Gorenstein, and hence flat over $H_{n}$. We derive two important consequences.

(1) We prove the strong form of the $n!$ conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients $K_{\lambda \mu }(q,t)$. This establishes the Macdonald positivity conjecture, namely that $K_{\lambda \mu }(q,t)\in {\mathbb N} [q,t]$.

(2) We show that the Hilbert scheme $H_{n}$ is isomorphic to the $G$-Hilbert scheme $(\mathbb{C}^{2})^{n}{//}S_n$ of Nakamura, in such a way that $X_{n}$ is identified with the universal family over $({\mathbb C}^2)^n{//}S_n$. From this point of view, $K_{\lambda \mu }(q,t)$ describes the fiber of a character sheaf $C_{\lambda }$ at a torus-fixed point of $({\mathbb C}^2)^n{//}S_n$corresponding to $\mu$.

The proofs rely on a study of certain subspace arrangements $Z(n,l)\subseteq (\mathbb{C}^{2})^{n+l}$, called polygraphs, whose coordinate rings $R(n,l)$ carry geometric information about $X_{n}$. The key result is that $R(n,l)$ is a free module over the polynomial ring in one set of coordinates on $(\mathbb{C}^{2})^{n}$. This is proven by an intricate inductive argument based on elementary commutative algebra.