Ian Grojnowski and Mark Haiman (preprint) proved a positivity conjecture for LLT polynomials that combined with the previous result implies the Macdonald positivity conjecture for Macdonald polynomials, and extended the definition of LLT polynomials to arbitrary finite root systems.
These conjectures are now proved; the hardest and final step was proving the positivity, which was done by Mark Haiman (2001), by proving the n! conjecture.
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The n! conjecture of Adriano Garsia and Mark Haiman states that for each partition μ of n the space
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