The \(n\)! conjecture and a vector bundle on the Hilbert scheme of \(n\) points in the plane (Q1978175)

One of the major open problems regarding the Macdonald polynomials \(J_{\mu}(x;q,t)\), \(\mu\) being a partition of \(n\), is the conjecture that, expanded in terms of certain modified Schur symmetric functions, the coefficients of \(J_{\mu}(x;q,t)\) are polynomials in the two parameters \(q,t\) with nonnegative integer coefficients. \textit{A. M. Garsia} and \textit{M. Haiman} [Proc. Natl. Acad. Sci. USA 90, No. 8, 3607-3610 (1993; Zbl 0831.05062)] introduced some bigraded \(S_n\)-module \({\mathbf H}_{\mu}\) and conjectured that the dimension of \({\mathbf H}_{\mu}\) is \(n!\) and this is known as the \(n!\) conjecture. It is known that the \(n!\) conjecture implies the Macdonald positivity conjecture. Many computer experimensts and some partial results support the validity of the \(n!\) conjecture. With the intention of proving the \(n!\) conjecture by induction \textit{F. Bergeron} and \textit{A. M. Garsia} [CRM Proc. Lect. Notes. 22, 1-52 (1999; Zbl 0947.20009)] formulated several conjectures concerning intersections of modules \({\mathbf M}_{\nu}\) for partitions lying below a given partition \(\mu\) of \(n+1\). The paper under review gives an explanation of these conjectures of Bergeron and Garsia in an algebraic-geometric setting, interpreting them in the context of the Hilbert scheme of \(n\) points in the plane. The author constructs a coherent sheaf \({\mathcal P}\) on the Hilbert scheme and shows that the \(n!\) conjecture is true if and only if \({\mathcal P}\) is a locally free sheaf, i.e. a vector bundle. Then the author studies the restriction of \({\mathcal P}\) to subvarieties isomorphic to \(\mathbb{P}^{k-1}\) contained in the Hilbert scheme and reduces the series of conjectures of Bergeron and Garsia to one conjecture on the structure of this vector bundle restricted to a projective space \(\mathbb{P}^k\) embedded in the Hilbert scheme. Finally the author reinterprets geometric statements combinatorially.