 
Summary: A combinatorial proof of Macdonald positivity
Sami Assaf
University of California, Berkeley
sassaf@math.upenn.edu
Banff International Research Station
September 10, 2007
S. Assaf  A combinatorial proof of Macdonald positivity p.1/20
Macdonald positivity
In 1988, Macdonald defined Hµ(x; q, t) and proved it to be
symmetric. Writing Hµ(x; q, t) = K,µ(q, t)s(x), he
conjectured:
Theorem. (Haiman 2001) We have K,µ(q, t) N[q, t].
Original proof realizes Hµ(x; q, t) as the Frobenius series of
doubly graded Snmodule (GarsiaHaiman module) using
algebraic geometry of the Hilbert scheme.
Newer proof due to Grojnowski and Haiman relates
Hµ(x; q, t) to LLT positivity (Haglund's formula) and then
uses KazhdanLusztig theory.
Problem: Find a combinatorial proof of positivity.
S. Assaf  A combinatorial proof of Macdonald positivity p.2/20
