3:00 p.m., Friday (Oct. 12)

Math 100

Mark Haiman

University of California, Berkeley

The geometric significance of Macdonald positivity

By classical results of Springer, Hotta, Lusztig and others, the symmetric functions known as Hall-Littlewood polynomials provide the solution to three closely related problems of geometry and representation theory: (1) describe the action of S_{n} on the cohomology H^{*}(X^{u}), where X=GL_{n}/B is the flag variety and u\in GL_{n} is a unipotent matrix; (2) describe the action of GL_{n}(q) on the finite flag variety X(q) in terms of irreducible GL_{n}(q) characters evaluated on unipotent matrices u\in GL_{n}(q); (3) describe the local intersection homology of the closure of a unipotent conjugagy class. The geometric interpretations imply that the ``q-Kostka coefficients'' K_{\lambda \mu }(q) relating Hall-Littlewood polynomials to Schur functions are polynomials with positive integer coefficients.

In 1988, I.G. Macdonald introduced generalized Hall-Littlewood polynomials, now known as Macdonald polynomials, which depend on two parameters q and t. He showed that the resulting ``q,t-Kostka coefficients'' K_{\lambda \mu }(q,t) enjoy a pleasant symmetry between the roles of q and t, and he conjectured that they too are polynomials with positive integer coefficients. Recently I proved Macdonald's positivity conjecture. The proof involves a representation-theoretic interpretation which A. Garsia and I conjectured in 1991, together with a geometric interpretation first suggested by C. Procesi, involving the Hilbert scheme of points in the plane.

The geometric interpretation of K_{\lambda \mu }(q,t) is different from the classical Springer/Lusztig interpretation of K_{\lambda \mu }(q). Hints of the link between the two are just emerging, along with clues to a possible extension to other Lie groups and Weyl groups.

Refreshments will be served in Math Annex Room 1115, 2:45 p.m.

Copyright © 2001 UBC Mathematics Department