Notes on MacDonald polynomials and the geometry of Hilbert schemes.

*(English)*Zbl 1057.14011
Fomin, S. (ed.), Symmetric functions 2001: Surveys of developments and perspectives. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, June 25–July 6, 2001. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0773-6/hbk). NATO Sci. Ser. II, Math. Phys. Chem. 74, 1-64 (2002).

From the text: These notes are based on series of seven lectures given in the combinatorics seminar at U.C. San Diego; in February and March, 2001. We discuss a series of new results in combinatorics, algebra and geometry. The main combinatorial problems we solve are

(1) we prove the positivity conjecture for Macdonald polynomials, and

(2) we prove a series of conjectures relating the diagonal harmonics to various familiar combinatorial enumerations; in particular we prove that the dimension of the space of diagonal harmonics is \((n+1)^{n-1}\).

In order to prove these results, we have to work out some new results about geometry of the Hilbert scheme of points in the plane and a certain related algebraic variety. As a technical tool for our geometric results, in turn, we need to do some commutative algebra, which although complicated, has a quite explicit and combinatorial nature.

For the entire collection see [Zbl 0997.00015].

(1) we prove the positivity conjecture for Macdonald polynomials, and

(2) we prove a series of conjectures relating the diagonal harmonics to various familiar combinatorial enumerations; in particular we prove that the dimension of the space of diagonal harmonics is \((n+1)^{n-1}\).

In order to prove these results, we have to work out some new results about geometry of the Hilbert scheme of points in the plane and a certain related algebraic variety. As a technical tool for our geometric results, in turn, we need to do some commutative algebra, which although complicated, has a quite explicit and combinatorial nature.

For the entire collection see [Zbl 0997.00015].