Hall-Littlewood polynomials, alcove walks, and the Macdonald polynomial inv statistic
Topics in Combinatorial Representation Theory
March 19, 2008 11:00 AM to 12:00 PM
Speakers:
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Abstract: |
A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund,
Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a
pair of statistics on fillings of Young diagrams. The inv statistic, which is the more
intricate one, suffices for specializing a closely related formula to one for the type A
Hall-Littlewood polynomials (spherical functions on p-adic groups). An apparently unrelated
development, at the level of arbitrary finite root systems, led to Schwer's formula (rephrased
and rederived by Ram) for the Hall-Littlewood polynomials of arbitrary type. The latter
formulas are in terms of so-called galleries/alcove walks, which originate in the work of
Gaussent-Littelmann and of myself with Postnikov on discrete versions of the Littelmann path
model. In this talk, I relate the above developments, by explaining how the inv statistic is
the outcome of "compressing" Ram's formula in type A. I also report on progress toward
deriving an analog of the inv statistic in types B and C in a similar manner, from Ram's
formula in these cases. |
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