Macdonald Polynomials and Geometry
University of California, San Diego
refreshments at 3:45pm
The theory of Macdonald polynomials (certain remarkable
symmetric functions) leads to difficult and profound positivity
conjectures, which remain unproven. The conjectured positive
quantities are expected to describe irreducible character
multiplicities in certain graded modules that Garsia and I constructed
some time ago. The natural setting for studying these modules is the
geometry of the Hilbert scheme of points in the plane. I will explain
the geometric conjectures which imply the Macdonald positivity
conjecture, and outline their connections with some other interesting
conjectures such as the McKay correspondence and the conjectured
Cohen-Macaulay property of the variety of pairs of commuting matrices.
Speaker's Contact Info: mhaiman(at-sign)macaulay.ucsd.edu
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