Current Developments in Mathematics

Combinatorics, symmetric functions, and Hilbert schemes

Mark Haiman

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Abstract

We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald's symmetric functions, and the "n!" and "(n+1)n-1" conjectures relating Macdonald polynomials to the characters of doubly-graded Sn modules. To make the treatment self-contained, we include background material from combinatorics, symmetric function theory, representation theory and geometry. At the end we discuss future directions, new conjectures and related work of Ginzburg, Kumar and Thomsen, Gordon, and Haglund and Loehr.

Article information

Source
Current Developments in Mathematics Volume 2002 (2002), 39-111.

Dates
First available in Project Euclid: 29 June 2004

Permanent link to this document
http://projecteuclid.org/euclid.cdm/1088530398

Mathematical Reviews number (MathSciNet)
MR2051783

Zentralblatt MATH identifier
1053.05118

Citation

Haiman , Mark. Combinatorics, symmetric functions, and Hilbert schemes. Current Developments in Mathematics 2002 (2002), 39--111. http://projecteuclid.org/euclid.cdm/1088530398.


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