Combinatorics, Symmetric Functions, and Hilbert Schemes

  title={Combinatorics, Symmetric Functions, and Hilbert Schemes},
  author={Mark D. Haiman},
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… CONTINUE READING

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